Ratios conjecture for quadratic twists of modular L-functions

Abstract

We develop the L-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of quadratic twist of modular L-functions using multiple Dirichlet series under the generalized Riemann hypothesis.

1 Introduction

As a significant conjecture with many important applications, the L-functions ratios conjecture makes predictions on the asymptotic behaviors of the sum of ratios of products of shifted L-functions. This conjecture originated from the work of D. W. Farmer [1] on shifted moments of the Riemann zeta function, and is formulated for general L-functions by J. B. Conrey, D. W. Farmer and M. R. Zirnbauer in [2, Section 5].

There are now several results available in the literature on the ratios conjecture, all valid for certain ranges of the relevant parameters, starting with the work of H. M. Bui, A. Florea and J. P. Keating in [3] over function fields for quadratic L-functions. Utilizing the powerful tool of multiple Dirichlet series, M. Čech [4] studied the case of quadratic Dirichlet L-functions under the assumption of the generalized Riemann hypothesis (GRH). This was the first result of its type over number fields.

Following the approach of M. Čech in [4], the authors studied the ratios conjecture for quadratic Hecke L-functions over the Gaussian field ${\mathbb {Q}}(i)$ in [5]. It is the aim of this paper to further apply the method of multiple Dirichlet series to develope the ratios conjecture for quadratic twists of modular L-functions. To state our result, we fix a holomorphic Hecke eigenform f of weight $\kappa$ of the full modular group $SL_2 ({\mathbb {Z}})$. The Fourier expansion of f at infinity can be written as

$$\begin{aligned} f(z) = \sum _{n=1}^{\infty } \lambda _f (n) n^{(\kappa -1)/2} e(nz), \quad \text{ where } \quad e(z)= \exp (2 \pi i z) . \end{aligned}$$

We reserve the letter p for a prime throughout the paper. For any Dirichlet character $\chi$ modulo d, the twisted modular L-function $L(s, f \otimes \chi )$ is defined for $\Re (s)>1$ by

$$\begin{aligned} L(s, f \otimes \chi )&= \sum _{n=1}^{\infty } \frac{\lambda _f(n)\chi (n)}{n^s} = \prod _{p\not \mid d} \left( 1 - \frac{\lambda _f (p) \chi (p)}{p^s} + \frac{1}{p^{2s}}\right) ^{-1}. \end{aligned}$$

(1.1)

We also write $L(s, \text {sym}^2f)$ for the symmetric square L-function of f defined in (2.9) and note that $L(s, \text {sym}^2f)$ is holomorphic for $\Re (s) \ge 1/2$ (see the discussions given in Sect. 2.1).

Let $\chi ^{(m)}=\left( \frac{m}{\cdot }\right)$ denote the Kronecker symbol defined on [6, p. 52] for any integer $m \equiv 0, 1 \pmod 4$. Note that every such m factors uniquely into $m=dl^2$, where d is a fundamental discriminant, i.e. d is square-free, $d \equiv 1 \pmod 4$ or $d=4n$ with n square-free, $n \equiv 2,3 \pmod 4$. Furthermore, when d is a fundamental discriminant, the function $L(s, f \otimes \chi ^{(d)})$ has an analytical continuation to the entire complex plane and satisfies the functional equation (see, for example, [7])

$$\begin{aligned} \Lambda (s, f \otimes \chi ^{(d)}) \!=:\! \left( \frac{|d|}{2\pi } \right) ^s \Gamma \left( s \!+\! \frac{\kappa -1}{2} \right) L(s, f \otimes \chi ^{(d)}) \!=\! i^\kappa \epsilon (d ) \Lambda (1- s, f \otimes \chi ^{(d)}),\qquad \end{aligned}$$

(1.2)

where $\epsilon (d) = 1$ if $d >0$ and $\epsilon (d) = -1$ if $d<0$.

For an odd, positive integer n, we denote $\chi _n$ for the quadratic character $\left( \frac{\cdot }{n} \right)$. Given any L-function, we write $L^{(c)}$ (resp. $L_{(c)}$) for the function given by the Euler product defining L but omitting those primes dividing (resp. not dividing) c. We also write $L_p$ for $L_{(p)}$. We observe that the quadratic reciprocity law implies that $L^{(2)}(s, f \otimes \chi _n)=L(s, f \otimes \chi ^{(4n)})$ when $n \equiv 1 \pmod 4$ and that $L^{(2)}(s, f \otimes \chi _n)=L(s, f \otimes \chi ^{(-4n)})$ when $n \equiv -1 \pmod 4$. It follows from (1.1), (1.2) and the above discussions that $L^{(2)}(s, f \otimes \chi _n)$ can also be continued analytically to the entirety of ${\mathbb {C}}$.

Our result in this paper investigates the ratios conjecture with one shift in the numerator and denominator for the family of quadratic twist of modular L-functions $L^{(2)}(s, f \otimes \chi _n)$ averaged over all odd, positive n.

Theorem 1.1

With the notation as above and the truth of GRH, let w(t) be a non-negative Schwartz function and ${\widehat{w}}(s)$ its Mellin transform. For any $\varepsilon >0$, $1/2>\Re (\alpha )>0$, $\Re (\beta )>\varepsilon$, we have

$$\begin{aligned} \begin{aligned} \sum _{\begin{array}{c} (n,2)=1 \end{array}}\frac{L^{(2)}(\tfrac{1}{2}+\alpha , f \otimes \chi _{n})}{L^{(2)}(\tfrac{1}{2}+\beta , f \otimes \chi _{n})}w \left( \frac{n}{X}\right)&= X\widehat{w}(1)L^{(2)}(1+2\alpha , \textrm{sym}^2 f)P(1,\tfrac{1}{2}+\alpha ,\tfrac{1}{2}+\beta ;f)\\&\quad +O\left( (1+|\alpha |)^{\varepsilon }|\beta |^\varepsilon X^{N(\alpha ,\beta )+\varepsilon }\right) , \end{aligned} \end{aligned}$$

(1.3)

where

$$\begin{aligned} \begin{aligned} P(s, w,z;f)&= \left( 1-\frac{1}{2^s} \right) \frac{1}{\zeta ^{(2)}(2w)}\\&\quad \times \prod _{p>2}\left( 1+\frac{1}{p^{2z}} \left( 1-\frac{1}{p^s} \right) \left( 1+\frac{1}{p^{2w}} \right) +\left( 1-\frac{1}{p^s} \right) \frac{1}{p^{2w}}\right. \\&\left. \quad -\frac{ \lambda ^2_f(p)-2}{p^{2w+s}}+\frac{1}{p^{4w+s}} -\left( 1-\frac{1}{p^s}\right) \frac{\lambda ^2_f(p)}{p^{z+w}} \right) , \end{aligned} \end{aligned}$$

(1.4)

and

$$\begin{aligned} N(\alpha ,\beta )=\max \left\{ 1-2\Re (\alpha ),1-2\Re (\beta ) \right\} . \end{aligned}$$

(1.5)

It follows from our discussions in Section 3.2 below that the value $P(1,\tfrac{1}{2}+\alpha ,\tfrac{1}{2}+\beta ;f)$ is finite for the ranges of $\alpha$ and $\beta$ defined in the statement of Theorem 1.1. The main term in (1.3) is consistent with the ratios conjecture, which can be derived following the treatments given in [8, Section 5]. However, the error term in (1.3) is inferior to the prediction of the ratios conjecture. The latter asserts that (1.3) holds uniformly for $|\Re (\alpha )|< 1/4$, $(\log X)^{-1} \ll \Re (\beta ) < 1/4$ and $\Im (\alpha ), \; \Im (\beta ) \ll X^{1-\varepsilon }$ with an error term $O(X^{1/2+\varepsilon })$. Nevertheless, the advantage of our result in (1.3) is that there is no constraint on imaginary parts of $\alpha$ or $\beta$.

Theorem 1.1 will be established using the method in the proof of [4, Theorem 1.2]. In particular, we note that the functional equation of a general (not necessarily primitive) quadratic Dirichlet L-function given in [4, Proposition 2.3] plays a crucial role in our proof.

2 Preliminaries

We first include some auxiliary results.

2.1 Quadratic Gauss sums

Recall that $\chi ^{(d)}=\left( \frac{d}{\cdot }\right)$ for the Kronecker symbol. Moreover, let $\psi _j, j \in \{ \pm 1, \pm 2\}$ denote the quadratic characters given by $\psi _j=\chi ^{(4j)}$. Note that $\psi _j$ is a primitive character modulo 4j for each j. We also denote by $\psi _0$, the primitive principal character, i.e. the trivial character modulo 1.

For any integer q and any Dirichlet character $\chi$ modulo n, we define the associated Gauss sum $\tau (\chi ,q)$ by

$$\begin{aligned} \tau (\chi ,q)=\sum _{j\pmod n}\chi (j)e \left( \frac{jq}{n} \right) . \end{aligned}$$

The following result is quoted from [4, Lemma 2.2].

Lemma 2.2

• (1) If $l\equiv 1 \pmod 4$, then

  $$\begin{aligned} \tau \left( \chi ^{(4l)},q\right) = {\left\{ \begin{array}{ll} 0,&{}\hbox {if }(q,2)=1,\\ -2\tau \left( \chi _l,q\right) ,&{}\hbox {if }q\equiv 2 \pmod 4,\\ 2\tau \left( \chi _l,q\right) ,&{}\hbox {if }q\equiv 0 \pmod 4. \end{array}\right. } \end{aligned}$$

• (2) If $l \equiv 3 \pmod 4$, then

  $$\begin{aligned} \tau \left( \chi ^{(4l)},q\right) ={\left\{ \begin{array}{ll} 0,&{}\hbox {if }2|q,\\ -2i\tau \left( \chi _l,q\right) ,&{}\hbox {if }q\equiv 1 \pmod 4,\\ 2i\tau \left( \chi _l,q\right) ,&{}\hbox {if }q\equiv 3 \pmod 4. \end{array}\right. } \end{aligned}$$

We further define $G\left( \chi _n,q\right)$ by

$$\begin{aligned} \begin{aligned} G\left( \chi _n,q\right)&=\left( \frac{1-i}{2}+\left( \frac{-1}{n}\right) \frac{1+i}{2}\right) \tau \left( \chi _n,q\right) ={\left\{ \begin{array}{ll} \tau \left( \chi _n,q\right) ,&{}\hbox {if }n\equiv 1 \pmod 4,\\ -i\tau \left( \chi _n,q\right) ,&{}\hbox {if }n\equiv 3\pmod 4. \end{array}\right. } \end{aligned} \end{aligned}$$

We denote $\varphi (m)$ for the Euler totient function of m. Our next result is taken from [9, Lemma 2.3] and evaluates $G\left( \chi _m,q\right)$.

Lemma 2.3

If $(m,n)=1$ then $G(\chi _{mn},q)=G(\chi _m,q)G(\chi _n,q)$. Suppose that $p^a$ is the largest power of p dividing q (put $a=\infty$ if $q=0$). Then for $k \ge 0$ we have

$$\begin{aligned} G\left( \chi _{p^k},q\right) ={\left\{ \begin{array}{ll}\varphi (p^k),&{}\hbox {if }k\le a, k\hbox { even,}\\ 0,&{}\hbox {if }k\le a, k\hbox { odd,}\\ -p^a,&{}\hbox {if }k=a+1, k\hbox { even,}\\ \left( \frac{qp^{-a}}{p}\right) p^{a}\sqrt{p},&{}\hbox {if }k=a+1, k\hbox { odd,}\\ 0,&{}\hbox {if }k\ge a+2. \end{array}\right. } \end{aligned}$$

2.2 Modular L-functions

For any Dirichlet character $\chi$, the twisted modular L-function $L(s, f \otimes \chi )$ has an Euler product for $\Re (s)>1$ given by

$$\begin{aligned} L(s, f \otimes \chi )&= \prod _{p} \prod ^2_{j =1} (1-\alpha _{f}(p,j)\chi (p)p^{-s})^{-1}. \end{aligned}$$

(2.1)

By Deligne’s proof [10] of the Weil conjecture, we know that

$$\begin{aligned} |\alpha _{f}(p,1)|=|\alpha _{f}(p,2)|=1 \quad \text{ and } \quad \alpha _{f}(p,1)\alpha _{f}(p,2)=1. \end{aligned}$$

(2.2)

It follows from this and (1.1) that $\lambda _f(n)$ is multiplicative and for $\nu \ge 1$, we have

$$\begin{aligned} \begin{aligned} \lambda _{f}(p^{\nu })&= \sum ^{\nu }_{j=0}\alpha ^{\nu -j}_f(p,1)\alpha ^j_{f}(p,2). \end{aligned} \end{aligned}$$

(2.3)

The above relation further implies that $\lambda _f (n) \in {\mathbb {R}}$, with the normalization $\lambda _f (1) =1$,

$$\begin{aligned} \begin{aligned} |\lambda _{f}(n)| \le d(n) \ll n^{\varepsilon }, \end{aligned} \end{aligned}$$

(2.4)

where d(n) is the number of divisors of n and the last estimate above follows from [11, Theorem 2.11].

We also deduce from (2.1) that for $\Re (s)>1$,

$$\begin{aligned} \begin{aligned} L^{-1}(s, f \otimes \chi )=\prod _p\prod ^2_{j=1}(1-\alpha _{f}(p, j)\chi (p)p^{-s})=:\sum ^{\infty }_{n=1}\frac{c_{f}(n)\chi (n)}{n^s}. \end{aligned} \end{aligned}$$

(2.5)

It follows from (2.2), (2.3) and the Euler product given in (2.5) that $c_{f}(n)$ is a multiplicative function of n such that

$$\begin{aligned} \begin{aligned} c_{f}(p^k)= \displaystyle {\left\{ \begin{array}{ll} 1, \quad k=0, 2, \\ -\lambda _f(p), \quad k=1, \\ 0, \quad k>2. \end{array}\right. } \end{aligned} \end{aligned}$$

(2.6)

In particular, we deduce from (2.2) and (2.6) that we have $|c_{f}(p^k)| \le 2$ for all $k \ge 0$, so that for any $\varepsilon >0$,

$$\begin{aligned} \begin{aligned} c_{f}(n) \ll 2^{\omega (n)} \ll n^{\varepsilon }, \end{aligned} \end{aligned}$$

(2.7)

where $\omega (n)$ denotes the number of distinct primes dividing n and the last estimation above follows from the well-known bound (see [11, Theorem 2.10])

$$\begin{aligned} \omega (h) \ll \frac{\log h}{\log \log h}, \; \text{ for } \; h \ge 3. \end{aligned}$$

(2.8)

Recall also that the symmetric square L-function $L(s, {\text {sym}}^2 f)$ of f is defined for $\Re (s)>1$ by (see [6, p. 137] and [6, (25.73)])

$$\begin{aligned} \begin{aligned} L(s, {\text {sym}}^2 f)&= \prod _p\prod _{1 \le i \le j \le 2} \left( 1-\frac{\alpha _{f}(p, i)\alpha _{f}(p, j)}{p^s} \right) ^{-1}\\&=\prod _p \left( 1-\frac{\alpha ^2_{f}(p, 1)}{p^s} \right) ^{-1} \left( 1-\frac{1}{p^s} \right) ^{-1} \left( 1-\frac{\alpha ^2_{f}(p, 2)}{p^s} \right) ^{-1} \\&= \zeta (2s) \sum _{n = 1}^{\infty } \frac{\lambda _f(n^2)}{n^s}=\prod _{p} \left( 1-\frac{\lambda _f(p^2)}{p^s}+\frac{\lambda _f(p^2)}{p^{2s}}-\frac{1}{p^{3s}} \right) ^{-1}. \end{aligned} \end{aligned}$$

(2.9)

It follows from a result of G. Shimura [12] that the corresponding completed L-function

$$\begin{aligned} \Lambda (s, {\text {sym}}^2 f)&= \pi ^{-3s/2}\Gamma \left( \frac{s+1}{2} \right) \Gamma \left( \frac{s+\kappa -1}{2} \right) \Gamma \left( \frac{s+\kappa }{2} \right) L(s, {\text {sym}}^2 f) \end{aligned}$$

(2.10)

is entire and satisfies the functional equation $\Lambda (s, {\text {sym}}^2 f)=\Lambda (1-s, {\text {sym}}^2 f)$.

2.3 Functional equations for Dirichlet L-functions

A key ingredient needed in our proof of Theorem 1.1 is the following functional equation for all Dirichlet characters $\chi$ modulo n from [4, Proposition 2.3].

Lemma 2.6

Let $\chi$ be any Dirichlet character modulo $n \ne \square$ such that $\chi (-1)=1$. Then we have

$$\begin{aligned} L(s,\chi )=\frac{\pi ^{s-1/2}}{n^s}\frac{\Gamma \left( \frac{1-s}{2}\right) }{\Gamma \left( \frac{s}{2}\right) } K(1-s,\chi ), \quad \text{ where } \quad K(s,\chi )=\sum _{q=1}^\infty \frac{\tau (\chi ,q)}{q^s}. \end{aligned}$$

(2.11)

2.4 Bounding L-functions

In this section, we gather various estimates on the values of L-functions under GRH, necessary in the sequal. For any quadratic character $\psi$ modulo n, we write ${\widehat{\psi }}$ for the primitive character that induces $\psi$. Note that every such ${{\widehat{\psi }}}$ can be written in form ${{\widehat{\psi }}}=\chi ^{(d)}$ for some fundamental discriminant d|n (see [11, Theorem 9.13]). We further write n uniquely as $n=n_1n_2$ with $(n_1, d)=1$ and $p |n_2 \Rightarrow p|d$. Using these notations, we get that for any integer q,

$$\begin{aligned} \begin{aligned} L^{(q)}( s, f \otimes \psi ) =L( s, f \otimes {\widehat{\psi }}) \prod _{p|qn_1} \left( 1-\frac{\alpha _f(p,1){{\widehat{\psi }}}(p)}{p^s} \right) \left( 1-\frac{\alpha _f(p,2){{\widehat{\psi }}}(p)}{p^s} \right) . \end{aligned} \end{aligned}$$

(2.12)

We now apply (2.2) to see that for $i=1,2$,

$$\begin{aligned} \Big |1-\frac{\alpha _f(p,i){{\widehat{\psi }}}(p)}{p^s}\Big | \le 2p^{\max (0,-\Re (s))}. \end{aligned}$$

It follows from the above that

$$\begin{aligned} \begin{aligned} \prod _{p|qn_1} \left( 1-\frac{\alpha _f(p,1){{\widehat{\psi }}}(p)}{p^s} \right) \left( 1-\frac{\alpha _f(p,2){{\widehat{\psi }}}(p)}{p^s} \right)&\ll 4^{\omega (q_1n)}(qn_1)^{\max (0,-\Re (2s))}\\&\ll (qn_1)^{\max (0,-\Re (2s))+\varepsilon }, \end{aligned} \end{aligned}$$

(2.13)

where the last estimate above follows from (2.8).

It follows from this and the functional equation in (1.2) that, for $\Re (s) \le 1/2$,

$$\begin{aligned} \begin{aligned} L( s, f \otimes {\widehat{\psi }})&\ll n^{1-2\Re (s)}\frac{\Gamma (1-s+ \frac{\kappa -1}{2})}{\Gamma (s+ \frac{\kappa -1}{2})}L( 1-s, f \otimes {\widehat{\psi }}) \\&\ll (n(1+|s|))^{1-2\Re (s)}L( 1-s, f \otimes {\widehat{\psi }}), \end{aligned} \end{aligned}$$

(2.14)

where the last bound above follows from Stirling’s formula (see [6, (5.113)]), which gives that for constants $c_0, d_0$,

$$\begin{aligned} \frac{\Gamma (c_0(1-s)+ d_0)}{\Gamma (c_0s+ d_0)} \ll (1+|s|)^{c_0(1-2\Re (s))}. \end{aligned}$$

(2.15)

We further note from [6, Theorem 5.19, Corollary 5.20] that under GRH, we have for $\Re (s) \ge 1/2$,

$$\begin{aligned} \begin{aligned}&\big | L\left( s, f \otimes {{\widehat{\psi }}} \right) \big |, \quad \big | L\left( s, {{\widehat{\psi }}} \right) \big | \ll |sn|^{\varepsilon }, \quad \big |L\left( s, \text {sym}^2f \right) \big | \ll |s|^{\varepsilon }. \end{aligned} \end{aligned}$$

(2.16)

The above combined with (2.12)–(2.16) now implies that

$$\begin{aligned} \begin{aligned} L^{(q)}( s, f \otimes \psi ) \ll (qn_1)^{\max (0,-\Re (s))+\varepsilon }(n(1+|s|))^{\max \{1-2\Re (s), 0\} +\varepsilon }. \end{aligned} \end{aligned}$$

(2.17)

Our discussions above also apply to other L-functions. For example, for the Dirichlet L-function $L(s, \psi )$. Under our notation above, if d is a fundamental discriminant, then we have the following functional equation (see [9, p. 456]) for $L(s, \chi ^{(d)})$.

$$\begin{aligned} \Lambda (s, \chi ^{(d)}) := \Big (\frac{|d|}{\pi } \Big )^{\frac{s}{2}}\Gamma \Big (\frac{s}{2} \Big )L(s, \chi ^{(d)})=\Lambda (1-s, \chi ^{(d)}). \end{aligned}$$

It follows from this and (2.15) that

$$\begin{aligned} \begin{aligned} L^{(q)}( s, \psi ) \ll (qn_1)^{\max (0,-\Re (s))+\varepsilon }(n(1+|s|))^{\max \{1/2-\Re (s),0 \} +\varepsilon }. \end{aligned} \end{aligned}$$

(2.18)

Similarly, by (2.10) and (2.15), we get

$$\begin{aligned} \begin{aligned} L^{(q)}(s, \text {sym}^2f ) \ll q^{\max (0,-\Re (s))+\varepsilon }(1+|s|)^{\max \{3(1/2-\Re (s)), 0\} +\varepsilon }. \end{aligned} \end{aligned}$$

(2.19)

Lastly, from [6, Theorem 5.19, Corollary 5.20], we have, under GRH, for $\Re (s) \ge 1/2+\varepsilon$,

$$\begin{aligned} \begin{aligned}&\big | L( s, f \otimes {{\widehat{\psi }}} ) \big |^{-1} \ll |sn|^{\varepsilon }. \end{aligned} \end{aligned}$$

(2.20)

2.5 Some results on multivariable complex functions

We include in this section some results from multivariable complex analysis. First we need the notation of a tube domain.

Definition 2.9

An open set $T\subset {\mathbb {C}}^n$ is a tube if there is an open set $U\subset {\mathbb {R}}^n$ such that $T=\{z\in {\mathbb {C}}^n:\ \Re (z)\in U\}.$

For a set $U\subset {\mathbb {R}}^n$, we define $T(U)=U+i{\mathbb {R}}^n\subset {\mathbb {C}}^n$. We have the following Bochner’s Tube Theorem [13].

Theorem 2.10

Let $U\subset {\mathbb {R}}^n$ be a connected open set and f(z) be a function holomorphic on T(U). Then f(z) has a holomorphic continuation to the convex hull of T(U).

The convex hull of an open set $T\subset {\mathbb {C}}^n$ is denoted by ${\widehat{T}}$. Then we quote the result from [4, Proposition C.5] on the modulus of holomorphic continuations of functions in multiple variables.

Proposition 2.11

Assume that $T\subset {\mathbb {C}}^n$ is a tube domain, $g,h:T\rightarrow {\mathbb {C}}$ are holomorphic functions, and let ${{\tilde{g}}},{{\tilde{h}}}$ be their holomorphic continuations to ${\widehat{T}}$. If $|g(z)|\le |h(z)|$ for all $z\in T$, and h(z) is nonzero in T, then also $|{{\tilde{g}}}(z)|\le |{{\tilde{h}}}(z)|$ for all $z\in {\widehat{T}}$.

3 Proof of Theorem 1.1

Let $\mu$ denote the Möbius function. Using the notations defined in Sect. 2.4, we define for $\Re (s), \Re (w), \Re (z)$ large enough,

$$\begin{aligned} \begin{aligned} A(s,w,z;f)&= \sum _{\begin{array}{c} (n,2)=1 \end{array}}\frac{L^{(2)}(w, f \otimes \chi _{n})}{L^{(2)}(z,f \otimes \chi _{n})n^s}\\&=\sum _{\begin{array}{c} (nmk,2)=1 \end{array}}\frac{\lambda _f(m)c_f(k)\chi _n(k)\chi _n(m)}{k^zm^wn^s}\\&= \sum _{\begin{array}{c} (mk,2)=1 \end{array}}\frac{\lambda _f(m)c_f(k)L\left( s,\chi ^{(4mk)} \right) }{m^wk^z}. \end{aligned} \end{aligned}$$

(3.1)

We shall devote the next few sections to articulating the analytical properties of A(s, w, z; f) as the proof of Theorem 1.1 relies crucially on them.

3.1 First region of absolute convergence of A(s, w, z; f)

We start with the series representation for A(s, w, z; f) given by the first equality in (3.1). This gives that if $\Re (z)>1/2$,

$$\begin{aligned} \begin{aligned} A(s,w,z;f)&= \sum _{\begin{array}{c} (n,2)=1 \end{array}}\frac{L^{(2)}(w,f \otimes \chi _n)}{L^{(2)}(z, f \otimes \chi _n)n^s}\\&= \sum _{\begin{array}{c} (h,2)=1 \end{array}}\frac{1}{h^{2s}}\mathop {{\mathop {\sum }\nolimits ^*}}\limits _{\begin{array}{c} (n,2)=1 \end{array}}\frac{L^{(2)}(w, f \otimes \chi _{n})\prod _{p | h}(1-\alpha _f(p,1)\chi _{n}(p)p^{-w})(1-\alpha _f(p,2)\chi _{n}(p)p^{-w}) }{n^s L^{(2)}(z, f \otimes \chi _{n})\prod _{p | h}(1-\alpha _f(p,1)\chi _{n}(p)p^{-z})(1-\alpha _f(p,2)\chi _{n}(p)p^{-z})} \\&\ll \sum _{\begin{array}{c} (h,2)=1 \end{array}}\frac{h^{\max (0,-2\Re (w))+\varepsilon } }{h^{2s}}\mathop {{\mathop {\sum }\nolimits ^*}}\limits _{\begin{array}{c} (n,2)=1 \end{array}}\Big | \frac{L^{(2)}(w, f \otimes \chi _n)}{L^{(2)}(z,f \otimes \chi _{n})n^s} \Big |, \end{aligned} \end{aligned}$$

(3.2)

where $\sum ^*$ henceforth denotes the sum over square-free integers and the last estimate above is obtained using (2.13) and a similar observation that if $\Re (z)>1/2$,

$$\begin{aligned} \begin{aligned} \prod _{p | h}(1-\alpha _f(p,1)\chi _{n}(p)p^{-z})^{-1}(1-\alpha _f(p,2)\chi _{n}(p)p^{-z})^{-1} \ll h^{\varepsilon }. \end{aligned} \end{aligned}$$

(3.3)

Recall that we have $L^{(2)}(w, f \otimes \chi _n)=L(w, f \otimes \chi ^{(\pm 4n)})$ for $n \equiv \pm 1 \pmod 4$. We write ${\widetilde{\chi }}_n$ for the primitive Dirichlet character that induces $\chi ^{(\pm 4n)}$. For a square-free n, we see that ${\widetilde{\chi }}_n=\chi ^{(n)}$ is a primitive character modulo n if $n \equiv 1 \pmod 4$ and ${\widetilde{\chi }}_n=\chi ^{(4n)}$ is a primitive character modulo 4n if $n \equiv -1 \pmod 4$. For $n \equiv 1 \pmod 4$, we have

$$\begin{aligned} \begin{aligned} \big |L^{(2)}(w, f \otimes \chi _n)\big |&= \big |(1-\alpha _f(2,1){\widetilde{\chi }}_n(2)2^{-w})(1- \alpha _f(2,2){\widetilde{\chi }}_n(2)2^{-w})L(w,f \otimes {\widetilde{\chi }}_n)\big |\\&\ll \big |L(w,f \otimes {\widetilde{\chi }}_n)\big |. \end{aligned} \end{aligned}$$

The above bound also holds for $n \equiv -1 \pmod 4$.

Similarly, if $\Re (z)>1/2$, we have, by (2.20) and an estimation analogous to (3.3), that under GRH,

$$\begin{aligned} \begin{aligned} |L^{(2)}(z, f \otimes \chi _n)|^{-1} \ll n^{\varepsilon }|L(z,f \otimes {\widetilde{\chi }}_n)|^{-1} \ll (|z|n)^{\varepsilon }. \end{aligned} \end{aligned}$$

The above, together with (3.2), allows us to deduce that if $\Re (z) >1/2$, then

$$\begin{aligned} \begin{aligned} A(s,w,z;f) \ll&\sum _{\begin{array}{c} (h,2)=1 \end{array}}\frac{h^{\max (0,-2\Re (w))+\varepsilon } }{h^{2s}}\mathop {{\mathop {\sum }\nolimits ^*}}\limits _{\begin{array}{c} (n,2)=1 \end{array}}\frac{|L(w, f \otimes {\widetilde{\chi }}_n)|}{|n^{s-\varepsilon }|}. \end{aligned} \end{aligned}$$

(3.4)

We now apply (2.17) to deduce from the above that under GRH, except for a simple pole at $w=1$, both sums of the right-hand side expression in (3.4) are convergent for $\Re (s)>1$, $\Re (w) \ge 1/2$, $\Re (z)>1/2$ as well as for $\Re (2s)>1$, $\Re (2s+2w)>1$, $\Re (s+2w)>2$, $\Re (w) < 1/2$, $\Re (z)>1/2$.

We thus conclude that the function A(s, w, z; f) converges absolutely in the region

$$\begin{aligned} S_0=\{(s,w,z): \Re (s)>1,\ \Re (2s+2w)>1,\ \Re (s+2w)>2,\ \Re (z)> \tfrac{1}{2} \}. \end{aligned}$$

Note that the condition $\Re (2s+2w)>1$ is implied by the other conditions so that we have

$$\begin{aligned} S_0=\{(s,w,z): \Re (s)>1, \ \Re (s+2w)>2,\ \Re (z)> \tfrac{1}{2} \}. \end{aligned}$$

Next, we deduce from the last expression of (3.1) that A(s, w, z; f) is given by the series

$$\begin{aligned} A(s,w,z;f)&=\sum _{\begin{array}{c} (mk,2)=1 \end{array}}\frac{\lambda _f(m)c_f(k) L( s, \chi ^{(4mk)})}{m^wk^z}. \end{aligned}$$

(3.5)

We write $mk=(mk)_0(mk)^2_1$ with $(mk)_0$ odd and square-free. Note that $\chi ^{((mk)_0)}$ is a primitive character modulo $(mk)_0$ when $(mk)_0 \equiv 1 \pmod 4$. We now apply (2.4), (2.7) and (2.18) to see that, except for a simple pole at $s=1$ arising from the summands with $mk=\square$, the sums over m, k such that $(mk)_0 \equiv 1 \pmod 4$ in (3.5) converge absolutely in the region

$$\begin{aligned} S_1&= \left\{ (s,w,z): \Re (w)>1,\ \Re (z)>1,\ \Re (s)\ge \tfrac{1}{2} \right\} \\&\quad \bigcup \left\{ (s,w,z): 0 \le \Re (s)< \tfrac{1}{2}, \ \Re (s+w)>\tfrac{3}{2},\ \Re (s+z)>\tfrac{3}{2} \right\} \\&\quad \bigcup \left\{ (s,w,z): \Re (s)<0, \ \Re (2s+w)>\tfrac{3}{2},\ \Re (2s+z)>\tfrac{3}{2} \right\} . \end{aligned}$$

Note that similar estimations hold when $(mk)_0 \equiv 2,3 \pmod 4$, in which case $\chi ^{(4(mk)_0)}$ is a primitive character modulo $4(mk)_0$. We thus conclude that, except for a simple pole at $s=1$ arising from the summands with $mk=\square$, the function A(s, w, z; f) converges absolutely in the region $S_1$.

To determine the convex hull of $S_0$ and $S_1$, we first note that for a fixed $\Re (z_0)>1$, the points $(1/2, 1, z_0)$ and $(1, 1/2, z_0)$ are in the closures of $S_0$ and $S_1$, respectively. These two points determine a line segment: $\Re (s+w)=3/2$ with $1/2 \le \Re (s) \le 1$ on the plane $\Re (z)=\Re (z_0)$. Note further that when $\Re (s)>1/2$ and $\Re (z)>1$, the conditions $\Re (s+z)>3/2$ and $\Re (2s+z)>3/2$ are automatically satisfied. We then deduce that the convex hull of $S_0$ and $S_1$ contains points (s, w, z) satisfying

$$\begin{aligned}{} & {} \{\Re (z)>1, \ \Re (s)>1/2, \ \Re (s+2w)>2,\ \Re (s+z)> \tfrac{3}{2}, \\{} & {} \quad \Re (s+w)>\tfrac{3}{2}, \ \Re (2s+z)>\tfrac{3}{2}\}. \end{aligned}$$

Combining the above set with the subsets of $S_1$ containing points with $\Re (s)<1/2$, we get that the convex hull of $S_0$ and $S_1$ contains points (s, w, z) satisfying

$$\begin{aligned} \{\Re (z)> & {} 1, \Re (s+2w)>2,\ \Re (s+z)> \tfrac{3}{2}, \ \nonumber \\ \Re (s+w)> & {} \tfrac{3}{2}, \ \Re (2s+w)>\tfrac{3}{2},\ \Re (2s+z)>\tfrac{3}{2}\}. \end{aligned}$$

(3.6)

On the other hand, if $1/2< \Re (z)< 1$, the points in $S_0$ are certainly contained in the convex hull of $S_0$ and $S_1$. We may thus focus on the case $\Re (s)<1$. In this case, we note that the points (1, 1/2, 1/2), (1, 1/2, 1) are in the closure of $S_0$ and (1/2, 1, 1) the closure of $S_1$. These three points determine a triangular region which can be regarded as the base of the region ${\mathcal {R}}$ enclosed by the four planes: $\Re (s+w)=3/2$, $\Re (s)=1$, $\Re (z)=1$, $\Re (s+z)=3/2$. It follows that the points in this triangle region are all in the convex hulls of $S_0$ and $S_1$. Further note that the points on the boundary ${\mathcal {R}} \cap \{\Re (z)=1\}$ of ${\mathcal {R}}$ are all in the convex hulls of $S_0$ and $S_1$ since they can be identified with the set $\{(s, w, z): \Re (z)=1, 1/2 \le \Re (s) \le 1, \Re (s+w) \ge 3/2\}$ and hence is contained in the convex hull of the set given in (3.6). Hence the entire region ${\mathcal {R}}$ lies in the convex hull of $S_0$ and $S_1$. Next, if $1/2< \Re (z)< 1$, the condition $\Re (s+z)>3/2$ implies that $\Re (s)>1/2$ so that one also has $\Re (2s+z)>3/2$. Similarly, when $1/2< \Re (z)< 1$, the condition $\Re (s+w)>3/2$ implies that $\Re (2s+w)>3/2$. Lastly, the condition $\Re (s+2w)>2$ implies $\Re (s+w)>3/2$ and $\Re (2\,s+w)>3/2$ for $\Re (s)>1$. It follows from the discussions here that the intersection of the convex hulls of $S_0$ and $S_1$ thus equals

$$\begin{aligned} S_2= & {} \{(s,w,z):\Re (z)> \tfrac{1}{2},\ \Re (s+2w)>2,\ \Re (s+z)\nonumber \\> & {} \tfrac{3}{2}, \ \Re (s+w)>\tfrac{3}{2}, \ \Re (2s+w)>\tfrac{3}{2},\ \Re (2s+z)>\tfrac{3}{2}\}. \end{aligned}$$

(3.7)

The above, together with Theorem 2.10, implies that $(s-1)(w-1)A(s,w,z;f)$ converges absolutely in the region $S_2$.

3.2 Residue of A(s, w, z; f) at $s=1$

It follows from (3.5) that A(s, w, z; f) has a pole at $s=1$ arising from the terms with $mk=\square$. To compute the corresponding residue and for our treatments later in the proof, we introduce the sum

$$\begin{aligned} \begin{aligned} A_1(s,w,z;f)&=: \sum _{\begin{array}{c} (mk,2)=1 \\ mk = \square \end{array}}\frac{\lambda _f(m)c_f(k)L\left( s, \chi ^{(4mk)}\right) }{m^wk^z} \\&= \sum _{\begin{array}{c} (mk,2)=1 \\ mk = \square \end{array}}\frac{\lambda _f(m)c_f(k)\zeta (s)\prod _{p | 2mk}(1-p^{-s}) }{m^wk^z} . \end{aligned} \end{aligned}$$

We further denote by $a_t(n)$ for any $t \in {\mathbb {C}}$ the multiplicative function such that $a_t(p^k)=1-1/p^t$ for any prime p. Thus, we recast $A_1(s,w,z;f)$ as

$$\begin{aligned} \begin{aligned} A_1(s,w,z;f) = \zeta ^{(2)}(s)\sum _{\begin{array}{c} (mk,2)=1 \\ mk = \square \end{array}}\frac{\lambda _f(m)c_f(k)a_s(mk) }{m^wk^z} . \end{aligned} \end{aligned}$$

We now write the last sum above as an Euler product. Slightly abusing notation by writing $p^{k'}$ with $k' \in {\mathbb {Z}}$ for the highest power of p dividing k, and similarly for m. Thus, we obtain, utilizing (2.6), that

$$\begin{aligned} \begin{aligned} A_1 (s,w,z;f)&= \zeta ^{(2)}(s)\prod _{p>2}\sum _{\begin{array}{c} m,k\ge 0\\ m+k\text { even} \end{array}}\frac{\lambda _f(p^m)c_f(p^k)a_s(p^{m+k})}{p^{mw+kz}}\\&= \zeta ^{(2)}(s)\prod _{p>2}\left( \sum _{\begin{array}{c} m\ge 0\\ 2 \mid m \end{array}}\frac{\lambda _f(p^m) a_s(p^m)}{p^{mw}}+\sum _{\begin{array}{c} m\ge 0\\ (m,2)=1 \end{array}}\frac{c_f(p)\lambda _f(p^m)a_s(p^{m+1})}{p^{z+mw}}\right. \\&\left. \quad +\sum _{\begin{array}{c} m\ge 0\\ 2 \mid m \end{array}}\frac{c_f(p^2)\lambda _f(p^m)a_s(p^{m+2})}{p^{2z+mw}}\right) \\&= \zeta ^{(2)}(s)\prod _{p>2}\left( 1+\left( 1-\frac{1}{p^s} \right) \frac{1}{p^{2z}}+\left( 1-\frac{1}{p^s} \right) \left( 1+\frac{1}{p^{2z}} \right) \sum ^{\infty }_{m=1}\frac{\lambda _f(p^{2m})}{p^{2mw}}\right. \\&\left. \quad -\left( 1-\frac{1}{p^s}\right) \frac{\lambda _f(p)}{p^z}\sum ^{\infty }_{m=0}\frac{\lambda _f(p^{2m+1})}{p^{(2m+1)w}} \right) . \end{aligned} \end{aligned}$$

(3.8)

We now use (2.3) to infer that (we may assume that $\alpha _f(p,1) \ne \alpha _f(p,2)$ as the other case follows from continuity)

$$\begin{aligned} \begin{aligned} \sum ^{\infty }_{i=0}\frac{\lambda _f(p^{2i})}{p^{iu}}&= \sum ^{\infty }_{i=0}\frac{1}{p^{iu}}\frac{\alpha ^{2i+1}_f(p,1)-\alpha ^{2i+1}_f(p,2)}{\alpha _f(p,1)-\alpha _f(p,2)}\\&= \left( 1-\frac{\alpha ^2_f(p,1)}{p^u} \right) ^{-1} \left( 1-\frac{\alpha ^2_f(p,2)}{p^u} \right) ^{-1} \left( 1+ \frac{1}{p^u} \right) . \end{aligned} \end{aligned}$$

(3.9)

Similarly, we have

$$\begin{aligned} \begin{aligned} \sum ^{\infty }_{i=0}\frac{\lambda _f(p^{2i+1})}{p^{(i+1)u}}&= \left( 1-\frac{\alpha ^2_f(p,1)}{p^u} \right) ^{-1} \left( 1-\frac{\alpha ^2_f(p,2)}{p^u} \right) ^{-1}\frac{\lambda _f(p)}{p^{u}}. \end{aligned} \end{aligned}$$

(3.10)

Inserting (3.9) and (3.10) into (3.8) yields

$$\begin{aligned} \begin{aligned} A_1(s,w,z;f)&= \zeta ^{(2)}(s)\prod _{p>2}\sum _{\begin{array}{c} m,k\ge 0\\ m+k\text { even} \end{array}}\frac{\lambda _f(p^m)c_f(p^k)a_s(p^{m+k})}{p^{mw+kz}} \\&= \zeta ^{(2)}(s)\prod _{p>2} \left( 1-\frac{\alpha ^2_f(p,1)}{p^{2w}} \right) ^{-1} \left( 1-\frac{\alpha ^2_f(p,2)}{p^{2w}} \right) ^{-1} \\&\quad \times \prod _{p>2}\left( \left( 1-\frac{1}{p^s} \right) \left( 1+\frac{1}{p^{2z}} \right) \left( 1+\frac{1}{p^{2w}} \right) \right. \\&\left. \quad +\frac{1}{p^s} \left( 1-\frac{\alpha ^2_f(p,1)}{p^{2w}} \right) \left( 1-\frac{\alpha ^2_f(p,2)}{p^{2w}} \right) -\left( 1-\frac{1}{p^s}\right) \frac{\lambda ^2_f(p)}{p^{z+w}} \right) \\&= \frac{\zeta ^{(2)}(s) L^{(2)}(2w, \text {sym}^2f)}{\zeta ^{(2)}(2w)} \\&\quad \times \prod _{p>2}\left( \left( 1-\frac{1}{p^s} \right) \left( 1+\frac{1}{p^{2z}} \right) \left( 1+\frac{1}{p^{2w}} \right) \right. \\&\left. \quad +\frac{1}{p^s} \left( 1-\frac{\alpha ^2_f(p,1)}{p^{2w}} \right) \left( 1-\frac{\alpha ^2_f(p,2)}{p^{2w}} \right) -\left( 1-\frac{1}{p^s}\right) \frac{\lambda ^2_f(p)}{p^{z+w}}\right) \\&= \frac{\zeta ^{(2)}(s) L^{(2)}(2w, \text {sym}^2f)}{\zeta ^{(2)}(2w)} \\&\quad \times \prod _{p>2}\left( 1+\frac{1}{p^{2z}} \left( 1-\frac{1}{p^s} \right) \left( 1+\frac{1}{p^{2w}} \right) +\left( 1-\frac{1}{p^s} \right) \frac{1}{p^{2w}}\right. \\&\left. \quad -\frac{ \alpha ^2_f(p,1)+\alpha ^2_f(p,2)}{p^{2w+s}}+\frac{1}{p^{4w+s}} -\left( 1-\frac{1}{p^s}\right) \frac{\lambda ^2_f(p)}{p^{z+w}} \right) \\&= \zeta (s)L^{(2)}(2w, \text {sym}^2f)P(s,w,z;f), \end{aligned} \end{aligned}$$

(3.11)

where P(s, w, z; f) is defined in (1.4).

We deduce from (1.4) and (3.11) that except for a simple pole at $s=1$, P(s, w, z; f) and $A_1(s,w,z;f)$ are holomorphic in the region

$$\begin{aligned} S_3&= \{(s,w,z):\ \Re (2z)>1, \ \Re (s+2z)>1, \nonumber \\&\quad \Re (s+2w)>1,\ \Re (w+z)>1, \ \Re (4w)>1, \ \Re (s+w+z)>1 \}. \end{aligned}$$

(3.12)

Recalling that the residue of $\zeta (s)$ at $s = 1$ is 1, we arrive at

$$\begin{aligned} \textrm{Res}_{s=1} A\left( s, \tfrac{1}{2}+\alpha ,\tfrac{1}{2}+\beta ;f\right)&= \textrm{Res}_{s=1} A_1\left( s, \tfrac{1}{2}+\alpha ,\tfrac{1}{2}+\beta ;f\right) \nonumber \\&=L^{(2)}(1+2\alpha , \text {sym}^2f)P\left( 1,\tfrac{1}{2}+\alpha ,\tfrac{1}{2}+\beta ;f\right) . \end{aligned}$$

(3.13)

3.3 Second region of absolute convergence of A(s, w, z; f)

From (3.5), we get

$$\begin{aligned} \begin{aligned} A(s,w,z;f)&= \sum _{\begin{array}{c} (mk,2)=1 \\ mk = \square \end{array}}\frac{\lambda _f(m)c_f(k)L\left( s, \chi ^{(4mk)}\right) }{m^wk^z} +\sum _{\begin{array}{c} (mk,2)=1 \\ mk \ne \square \end{array}}\frac{\lambda _f(m)c_f(k)L\left( s, \chi ^{(4mk)}\right) }{m^wk^z} \\&= \sum _{\begin{array}{c} (mk,2)=1 \\ mk = \square \end{array}}\frac{\lambda _f(m)c_f(k)\zeta (s)\prod _{p | 2mk}(1-p^{-s}) }{m^wk^z} +\sum _{\begin{array}{c} (mk,2)=1 \\ mk \ne \square \end{array}}\frac{\lambda _f(m)c_f(k)L\left( s, \chi ^{(4mk)}\right) }{m^wk^z} \\&=: A_1(s,w,z;f)+A_2(s,w,z;f). \end{aligned}\nonumber \\ \end{aligned}$$

(3.14)

We recall from our discussions in the previous section that except for a simple pole at $s=1$, $A_1(s,w,z;f)$ is holomorphic in the region $S_3$.

Next, we apply the functional equation given in Lemma 2.6 for $L\left( s, \chi ^{(4mk)}\right)$ in the case $mk \ne \square$ by observing that $\chi ^{(4mk)}$ is a Dirichlet character modulo 4mk for any $m,k\ge 1$ with $\chi ^{(4mk)}(-1)=1$. We obtain from (2.11) that

$$\begin{aligned} \begin{aligned} A_2(s,w,z;f) =\frac{\pi ^{s-1/2}}{4^s}\frac{\Gamma (\frac{1-s}{2})}{\Gamma (\frac{s}{2}) } C(1-s,s+w,s+z;f), \end{aligned} \end{aligned}$$

(3.15)

where C(s, w, z; f) is given by the triple Dirichlet series

$$\begin{aligned} C(s,w,z;f)&= \sum _{\begin{array}{c} q, m, k \\ (mk,2)=1 \\ mk \ne \square \end{array}}\frac{\lambda _f(m)c_f(k)\tau (\chi ^{(4mk)}, q)}{q^sm^wk^z} \\&= \sum _{\begin{array}{c} q, m, k \\ \begin{array}{c} (mk,2)=1 \end{array} \end{array}}\frac{\lambda _f(m)c_f(k)\tau (\chi ^{(4mk)}, q)}{q^sm^wk^z}-\sum _{\begin{array}{c} q, m, k \\ (mk,2)=1 \\ mk = \square \end{array}}\frac{\lambda _f(m)c_f(k)\tau (\chi ^{(4mk)}, q)}{q^sm^wk^z}. \end{aligned}$$

By (3.7), (3.12) and the functional equation (3.15), we see that C(s, w, z; f) is initially defined for $\Re (s)$, $\Re (z)$ and $\Re (w)$ sufficiently large. To extend this region, we exchange the summations in C(s, w, z; f) and set $mk=l$ to obtain that

$$\begin{aligned} \begin{aligned} C(s,w,z;f)&= \sum ^{\infty }_{q =1}\frac{1}{q^s}\sum _{\begin{array}{c} (l,2)=1 \end{array}}\frac{\tau \left( \chi ^{(4l)}, q \right) r(l, z-w)}{l^w}\\&\quad -\sum ^{\infty }_{q =1}\frac{1}{q^s}\sum _{\begin{array}{c} (l,2)=1 \\ l = \square \end{array}}\frac{\tau \left( \chi ^{(4l)}, q \right) r(l, z-w)}{l^w}\\&=: C_1(s,w,z;f)-C_2(s,w,z;f), \end{aligned} \end{aligned}$$

(3.16)

where

$$\begin{aligned} \begin{aligned} r(l, t)=\sum _{\begin{array}{c} k|l \end{array}}\frac{\lambda _f(l/k)c_f(k)}{k^{t}}. \end{aligned} \end{aligned}$$

(3.17)

We now define, for two Dirichlet characters $\psi$ and $\psi '$ whose conductors divide 8,

$$\begin{aligned} \begin{aligned} C_1(s,w,z;\psi ,\psi ',f)&=: \sum _{l,q\ge 1}\frac{G\left( \chi _l,q\right) \psi (l)\psi '(q) r(l, z-w)}{l^wq^s}, \quad \text{ and } \\ C_2(s,w,z;\psi ,\psi ',f)&=: \sum _{l,q\ge 1}\frac{G \left( \chi _{l^2},q\right) \psi (l)\psi '(q) r(l^2, z-w)}{l^{2w}q^s}. \end{aligned} \end{aligned}$$

(3.18)

Following the arguments contained in [4, §6.4] and making use of Lemma 2.2, we see that

$$\begin{aligned} \begin{aligned} C_1(s,w,z;f)&= -2^{-s}\big ( C_1(s,w,z;\psi _2,\psi _1,f)+C_1(s,w,z;\psi _{-2},\psi _1,f)\big ) \\&\quad +4^{-s}\big ( C_1(s,w,z;\psi _1,\psi _0,f) \\&\quad +C_1(s,w,z;\psi _{-1},\psi _0,f)\big ) +C_1(s,w,z;\psi _1,\psi _{-1},f)\\&\quad -C_1(s,w,z;\psi _{-1},\psi _{-1},f), \quad \text{ and } \\ C_2(s,w,z;f)&= -2^{1-s}C_2(s,w,z;\psi _1,\psi _1,f)+2^{1-2s}C_2(s,w,z;\psi _1,\psi _0,f). \end{aligned} \end{aligned}$$

(3.19)

Note that every integer $q \ge 1$ can be written uniquely as $q=q_1q^2_2$ with $q_1$ square-free. We may thus write

$$\begin{aligned} C_i(s,w,z;\psi ,\psi ',f)=\mathop {{\mathop {\sum }\nolimits ^*}}\limits _{q_1}\frac{\psi '(q_1)}{q_1^s}\cdot D_i(s, w,z-w;q_1, \psi ,\psi ',f), \quad i =1,2, \end{aligned}$$

(3.20)

where

$$\begin{aligned} \begin{aligned} D_1(s, w,t;q_1,\psi ,\psi ',f)&=\sum _{l,q_2=1}^\infty \frac{G\left( \chi _{l},q_1q^2_2\right) \psi (l)\psi '(q^2_2) r(l, t)}{l^wq^{2s}_{2}}, \quad \text{ and } \\ D_2(s, w,t;q_1,\psi ,\psi ',f)&=\sum _{l,q_2=1}^\infty \frac{G\left( \chi _{l^2},q_1q^2_2\right) \psi (l)\psi '(q^2_2) r(l^2, t)}{l^{2w}q^{2s}_{2}}. \end{aligned} \end{aligned}$$

(3.21)

We have the following result for the analytic properties of $D_i(s, w,t; q_1, \psi , \psi ', f)$.

Lemma 3.4

With the notation as above, for $\psi \ne \psi _0$, the functions $D_i(s, w,t; q_1,\psi ,\psi ',f), i=1,2$ have meromorphic continuations to the region

$$\begin{aligned} \{(s,w,t): \Re (s)>1, \ \Re (w)>1,\ \Re (w+t)>1\}. \end{aligned}$$

For $\Re (s)>1+\varepsilon , \Re (w)>1+\varepsilon$ and $\Re (w+t)>1+\varepsilon$, we have

$$\begin{aligned} |D_i(s, w,t;q_1, \psi ,\psi ', f)|\ll |q_1w(t+w)|^{\varepsilon }. \end{aligned}$$

(3.22)

Proof

As the proofs are similar, we consider only the case for $D_1(s, w,t;q_1, \psi ,\psi ',f)$ here. First $D_1(s,w,t;q_1, \psi ,\psi ',f)$ are jointly multiplicative functions of $l,q_2$ by Lemma 2.3 in the double sum defining $D_1$ in (3.21). Moreover, as $\psi \ne \psi _0$, we may assume that l is odd. We write $D_1(s,w,z;q_1, \psi ,\psi ',f)$ using (2.6) into an Euler product so that

$$\begin{aligned} \begin{aligned}&D_1(s, w,t;q_1, \psi ,\psi ',f)= \prod _p D_{1,p}(s, w,t;q_1, \psi ,\psi ',f). \end{aligned} \end{aligned}$$

(3.23)

Then we have

$$\begin{aligned} \begin{aligned}&D_{1,p}(s, w,t; q_1, \psi ,\psi ',f)= \displaystyle {\left\{ \begin{array}{ll} \displaystyle \sum _{l=0}^\infty \frac{ \psi '(2^{2l})}{2^{2ls}}, &{} p=2, \\ \displaystyle \sum _{l,k=0}^\infty \frac{ \psi (p^l)\psi '(p^{2k})G\left( \chi _{p^l}, q_1p^{2k} \right) r(p^l, t) }{p^{lw+2ks}}, &{} p>2. \end{array}\right. } \end{aligned} \end{aligned}$$

(3.24)

Note that for a fixed $p > 2$,

$$\begin{aligned} \begin{aligned}&\sum _{l,k=0}^\infty \frac{ \psi (p^l)\psi '(p^{2k})G\left( \chi _{p^l}, q_1p^{2k} \right) r(p^l, t)}{p^{lw+2ks}} \\&\quad = \sum _{l=0}^\infty \frac{ \psi (p^l)G\left( \chi _{p^l}, q_1 \right) r(p^l, t)}{p^{lw}} + \sum _{l \ge 0, k \ge 1}\frac{ \psi (p^l)\psi '(p^{2k})G\left( \chi _{p^l}, q_1p^{2k} \right) r(p^l, t)}{p^{lw+2ks}}. \end{aligned}\nonumber \\ \end{aligned}$$

(3.25)

Observe that

$$\begin{aligned} \begin{aligned} r(p^l, t)=\lambda _f(p^l)-\frac{\lambda _f(p^{l-1})\lambda _f(p)}{p^{t}}+\frac{\lambda _f(p^{l-2})}{p^{2t}}, \end{aligned} \end{aligned}$$

(3.26)

where we denote $\lambda _f(p^{-i})=0$ for integers $i \ge 0$.

Notice that by (2.2) and (2.3), we have

$$\begin{aligned} \begin{aligned} |\lambda _f(p^l)| \le l+1, \quad l \ge 0. \end{aligned} \end{aligned}$$

(3.27)

It follows from this and (3.26) that for $l \ge 1$,

$$\begin{aligned} \begin{aligned} |r(p^l, t)| \le \displaystyle {\left\{ \begin{array}{ll} 4(l+1)(1+p^{-t}), &{} l=1, \\ 4(l+1)(1+p^{-2t}), &{} l \ge 2. \end{array}\right. } \end{aligned} \end{aligned}$$

Also, note that Lemma 2.3 implies that

$$\begin{aligned} \begin{aligned} |G( \chi _{p^l}, q_1p^{2k} )| \ll p^l. \end{aligned} \end{aligned}$$

We apply the above estimations to see that when $\Re (s)> 1/2$, $\Re (w)>1$, $\Re (w+t)>1$,

$$\begin{aligned} \begin{aligned}&\sum _{l \ge 0, k \ge 1} \frac{ \psi (p^l)\psi '(p^{2k})G\left( \chi _{p^l}, q_1p^{2k} \right) r(p^l, t) }{p^{lw+2ks}}\\&\quad = \sum _{k \ge 1}\frac{ \psi '(p^{2k})G\left( \chi _{1}, q_1p^{2k} \right) }{p^{2ks}}\\&\qquad +\sum _{l, k \ge 1}\frac{ \psi (p^l)\psi '(p^{2k})G\left( \chi _{p^l}, q_1p^{2k} \right) r(p^l, t)}{p^{lw+2ks}} \\&\quad \ll p^{-2s} + p^{-2s} \left( \frac{1}{p^{w-1}}4(l+1)(1+p^{-t})+\sum _{l \ge 2}\frac{1}{p^{l(w-1)}}4(l+1)(1+p^{-2t}) \right) \\&\quad \ll p^{-2s}+p^{-2s-w+1}+p^{-2s-2w+2}+p^{-2s-w-t+1}+p^{-2s-2w-2t+2}\\&\quad \ll p^{-2s}+p^{-2s-w-t+1}. \end{aligned}\nonumber \\ \end{aligned}$$

(3.28)

We now apply Lemma 2.3 and (3.26) to see that when $p \not \mid 2q_1$ and $\Re (w)>1$,

$$\begin{aligned} \begin{aligned}&\sum _{l=0}^\infty \frac{ \psi (p^l)G\left( \chi _{p^l}, q_1 \right) r(p^l, t)}{p^{lw}} = 1+\frac{ \psi (p)\lambda _f(p)\chi ^{(q_1)}(p)}{p^{w-1/2}}\left( 1-\frac{1}{p^{t}}\right) \\&\quad = L_p \left( w-\tfrac{1}{2}, f \otimes \chi ^{(q_1)}\psi \right) \left( 1-\frac{\lambda _f^2(p)}{p^{2w-1}}-\frac{\chi ^{(q_1)}(p)\psi (p)\lambda _f(p)}{p^{w+t-1/2}}+\frac{\lambda _f^2(p)}{p^{2w+t-1}}\right) \\&\quad = \frac{L_{p}\left( w-\tfrac{1}{2},f \otimes \chi ^{(q_1)}\psi \right) }{L_{p}(2w-1, \text {sym}^2f)\zeta _{p}(2w-1)} \\&\qquad \times \left( 1-\frac{\chi ^{(q_1)}(p) \psi (p)\lambda _f(p)}{p^{w+t-1/2}}+O \Big (\frac{1}{p^{2w+t-1}}+\frac{1}{p^{w+t-1/2+2w-1}}+\frac{1}{p^{2(2w-1)}}\Big ) \right) \\&\quad = \frac{L_{p}\left( w-\tfrac{1}{2}, f \otimes \chi ^{(q_1)}\psi \right) }{L_{p}(2w-1, \text {sym}^2f)\zeta _{p}(2w-1)L_{p}\left( t+w-1/2,f \otimes \chi ^{(q_1)}\psi \right) } \\&\qquad \times \left( 1+O \Big (\frac{1}{p^{2w+2t-1}}+\frac{1}{p^{2w+t-1}}+\frac{1}{p^{w+t-1/2+2w-1}}+\frac{1}{p^{2(2w-1)}} \Big ) \right) . \end{aligned}\nonumber \\ \end{aligned}$$

(3.29)

We deduce from (3.24), (3.25), (3.28) and (3.29) that for $p \not \mid 2q_1$, $\Re (s)>\frac{1}{2}, \Re (w)>1, \Re (w+t)>1$,

$$\begin{aligned} \begin{aligned}&D_{1,p} (s, w, t;q_1, \psi ,\psi ',f) \\&\quad = \frac{L_{p}\left( w-\tfrac{1}{2}, f \otimes \chi ^{(q_1)}\psi \right) }{L_{p}(2w-1, \text {sym}^2f)\zeta _{p}(2w-1)L_{p}\left( t+w-1/2,f \otimes \chi ^{(q_1)}\psi \right) } \\&\qquad \times \Big ( 1+O \Big (p^{-(2w+2t-1)}+p^{-(2w+t-1)}+p^{-(w+t-1/2+2w-1)}\\&\qquad +p^{-2(2w-1)}+ p^{-2s}+p^{-2s-w-t+1}\Big ) \Big ) . \end{aligned} \end{aligned}$$

(3.30)

The first assertion of the lemma now follows from (3.23), (3.24) and (3.30).

We next note that Lemma 2.3, (3.17) and (3.27) implies that when $p | q_1, p \ne 2$,

$$\begin{aligned} \begin{aligned} \sum _{l=0}^\infty \frac{ \psi (p^l)G\left( \chi _{p^l}, q_1 \right) r(p^l, t)}{p^{lw}}&= 1-\frac{ \psi (p^2)}{p^{2w-1}}\left( \lambda _f(p^2)-\frac{\lambda ^2_f(p)}{p^{t}}+\frac{1}{p^{2t}}\right) \\&= 1+O(p^{-2w-1}+p^{-2w-2t-1}). \end{aligned} \end{aligned}$$

(3.31)

We thus deduce from (3.24), (3.28) and (3.31) that for $p | q_1$, $p \ne 2$, $\Re (s)>1/2$, $\Re (w)>1$, $\Re (w+t)>1$,

$$\begin{aligned} \begin{aligned}&D_{1,p}(s, w,t; q_1, \psi ,\psi ',f) = 1+O \Big (p^{-2w-1}+p^{-2w-2t-1}+ p^{-2s}+p^{-2s-w-t+1}\Big ) . \end{aligned} \end{aligned}$$

(3.32)

We conclude from (3.23), (3.24), (3.30) and (3.32) that for $\Re (s)>1+\varepsilon$, $\Re (w)>1+\varepsilon$ and $\Re (w+t)>1+\varepsilon$,

$$\begin{aligned} \begin{aligned} D_{1}(s, w,t;q_1, \psi ,\psi ',f)&\ll q_1^{\varepsilon }\Big |\frac{L^{(2q_1)}\left( w-\tfrac{1}{2}, f \otimes \chi ^{(q_1)}\psi \right) }{L^{(2q_1)}(2w-1, \text {sym}^2f)\zeta ^{(2q_1)}(2w-1)L^{(2q_1)}\left( t+w-1/2,f \otimes \chi ^{(q_1)}\psi \right) } \Big | \\&\ll q_1^{\varepsilon }\Big |\frac{L\left( w-\tfrac{1}{2}, f \otimes \chi ^{(q_1)}\psi \right) }{L(2w-1, \text {sym}^2f)\zeta (2w-1)L\left( t+w-1/2,f \otimes \chi ^{(q_1)}\psi \right) } \Big | \\&\ll |q_1w(w+t)|^{\varepsilon }, \end{aligned} \end{aligned}$$

where the last estimation above follows (2.16). This implies (3.22) and hence completes the proof of the lemma. $\square$

The above lemma now allows us to extend C(s, w, z; f) to the region

$$\begin{aligned} \{(s,w,z):\ \Re (s)>1, \ \Re (w)>1,\ \Re (z)>1\}. \end{aligned}$$

Using (3.12), (3.14) and the above, we can extend $(s-1)(w-1)A(s,w,z;f)$ to the region

$$\begin{aligned} S_4&= \{(s,w,z):\Re (2z)>1, \ \Re (s+2z)>1, \ \Re (s+2w)>1,\\&\quad \Re (w+z)>1, \ \Re (4w)>1, \ \Re (s+w+z)>1, \\&\quad \Re (s+w)>1,\ \Re (s+z)>1, \ \Re (1-s)>1\}. \end{aligned}$$

Note that the condition $\Re (1-s)>1$ is equivalent to $\Re (s)<0$ so that the conditions $\Re (s+w)>1, \Re (s+z)>1$ is the same as $\Re (w)>1$, $\Re (z)>1$. It follows that the rest of the conditions given in the definition of $S_4$ are superseded by the above three conditions. Thus

$$\begin{aligned} S_4=\{(s,w,z):\ \Re (s)<0, \ \Re (s+w)>1,\ \Re (s+z)>1\}. \end{aligned}$$

We further note that the region $S_2$ contains the subset given by

$$\begin{aligned} \{(s,w,z): \Re (z)>1, \ \Re (s)>1, \ \Re (s+2w)>2 \}. \end{aligned}$$

As the region $S_4$ contains points (s, w, z) such that

$$\begin{aligned} \{ \Re (s)<0, \ 1< \Re (s+w)<\Re (z), \ \Re (s)>1-\Re (z) \}, \end{aligned}$$

it is then readily seen that the convex hull of the above regions contains $S_5 \cap \{ (s,w,z): \Re (z)>1\}$, where

$$\begin{aligned} S_5&=\{(s,w,z):\ \qquad \quad \Re (s+2w)>2,\ \Re (s+2z)>2, \ \Re (s+z)>1, \\&\quad \Re (s+w)>1, \ \Re (w)> \tfrac{1}{4}, \ \Re (z)>\tfrac{1}{2} \}. \end{aligned}$$

On the other hand, when $1/2< \Re (z)< 1$, we note that the points in $S_2$ are certainly contained in the convex hull of $S_2$ and $S_4$ and one checks $S_2 \cap \{ (s,w,z): \Re (s)>1\}=S_5 \cap \{ (s,w,z): \Re (s)>1\}$. We may thus focus on the case $\Re (s)<1$. In this case, we note that the points (1, 1/2, 1/2), (1, 1/2, 1) are in the closure of $S_2$ and the point (0, 1, 1) is in the closure of $S_4$. These three points determine a triangular region which can be regarded as the base of the region ${\mathcal {S}}$ enclosed by the four planes: $\Re (s+2w)=2$, $\Re (s)=1$, $\Re (z)=1$, $\Re (s+2z)=2$. It follows that the points in this triangular region are all in the convex hulls of $S_2$ and $S_4$. Further note that the points on the boundary ${\mathcal {S}} \cap \{\Re (z)=1\}$ of ${\mathcal {S}}$ are all in the convex hull of $S_2$ and $S_4$ since they can be identified with the set $\{(s, w, z): \Re (z)=1, 0 \le \Re (s) \le 1, \Re (s+2w) \ge 2\}$ and hence is contained in the convex hull of $S_5 \cap \{ (s,w,z): \Re (z)>1\}$. We then deduce that the entire region ${\mathcal {S}}$ lies in the convex hull of $S_0$ and $S_1$. We next note that when $1/2< \Re (z)< 1$, the condition $\Re (s+2z)>2$ implies that $\Re (s)>0$ so that one also has $\Re (s+z)>1$ and that the condition $\Re (s+2w)>2$ implies $\Re (s+w)>1$. It follows from these discussions we see that the intersection of the convex hull of $S_2$ and $S_4$ thus contains $S_5$.

We apply Theorem 2.10 again to conclude that $(s-1)(w-1)A(s,w,z;f)$ converges absolutely in the region $S_5$.

3.4 Bounding A(s, w, z; f) in vertical strips

In order to prove Theorem 1.1, we also need to estimate |A(s, w, z; f)| in vertical strips.

We set for any fixed $0<\delta <1/1000$ and the previously defined regions $S_j$,

$$\begin{aligned} {\widetilde{S}}_j=S_{j,\delta }\cap \{(s,w,z):\Re (s)>-5/2,\ \Re (w)>1/2-\delta \}, \end{aligned}$$

where $S_{j,\delta }= \{ (s,w,z)+\delta (1,1,1) : (s,w,z) \in S_j \}$. Set

$$\begin{aligned} p(s,w)=(s-1)(w-1), \end{aligned}$$

so that p(s, w)A(s, w, z; f) is an analytic function in the regions under our consideration. We also write ${{\tilde{p}}}(s,w)=1+|p(s,w)|$.

We consider the bound for A(s, w, z; f) given in (3.4) and apply (2.17) to deduce that, under GRH, in ${\widetilde{S}}_0$,

$$\begin{aligned} \begin{aligned} |p(s,w)A(s,w,z;f)| \ll {{\tilde{p}}}(s,w)|wz|^{\varepsilon }(1+|w|)^{\max \{1-2\Re (w), 0 \}+\varepsilon }. \end{aligned} \end{aligned}$$

Similarly, using the estimates (2.18) in (3.5) renders that, under GRH, in the region ${\widetilde{S}}_1$,

$$\begin{aligned} |p(s,w)A(s,w,z;f)|\ll {{\tilde{p}}}(s,w)(1+|s|)^{\max \{1/2-\Re (s), 0\}+\varepsilon }. \end{aligned}$$

We then deduce from the above and Proposition 2.11 that in the convex hull ${\widetilde{S}}_2$ of ${\widetilde{S}}_0$ and ${\widetilde{S}}_1$, we have under GRH,

$$\begin{aligned} |p(s,w)A(s,w,z;f)|\ll {{\tilde{p}}}(s,w) |wz|^{\varepsilon }(1+|w|)^{\max \{1-2\Re (w),0 \}+\varepsilon }(1+|s|)^{3+\varepsilon }. \end{aligned}$$

(3.33)

Moreover, by (3.11) and the estimations given in (2.18) for $\zeta (s)$ (corresponding to the case $\psi =\psi _0$ being the primitive principal character) and in (2.19) for $L^{(2)}(2w, \text {sym}^2f)$ that in the region ${\widetilde{S}}_3$, under GRH,

$$\begin{aligned} |A_1(s,w,z;f)| \ll |w|^{\varepsilon }(1+|s|)^{\max \{(1-\Re (s))/2, 1/2-\Re (s), 0\}+\varepsilon }. \end{aligned}$$

(3.34)

Also, by (3.16)–(3.21) and Lemma 3.4,

$$\begin{aligned} |C(s,w,z;f)|\ll |wz|^{\varepsilon } \end{aligned}$$

(3.35)

in the region

$$\begin{aligned} \{(s,w,z):\Re (w)>1+\varepsilon ,\ \Re (z)>1+\varepsilon ,\ \Re (s) >1+\varepsilon \}. \end{aligned}$$

Now, applying (3.14), the functional equation (3.15), the bounds given in (3.34), (3.35), together with (2.15), we obtain that in the region ${\widetilde{S}}_4$,

$$\begin{aligned} |p(s,w)A(s,w,z;f)|\ll {{\tilde{p}}}(s,w) |wz|^{\varepsilon }(1+|w|)^{\max \{3(1/2-\Re (w)), 0\} +\varepsilon }(1+|s|)^{3+\varepsilon }. \end{aligned}$$

(3.36)

We now conclude from (3.33), (3.36) and Proposition 2.11 that in the convex hull ${\widetilde{S}}_5$ of ${\widetilde{S}}_2$ and ${\widetilde{S}}_4$,

$$\begin{aligned} |p(s,w)A(s,w,z;f)|\ll {{\tilde{p}}}(s,w)|wz|^{\varepsilon }(1+|w|)^{\max \{3(1/2-\Re (w)), 0\} +\varepsilon }(1+|s|)^{3+\varepsilon }. \end{aligned}$$

(3.37)

3.5 Completion of proof

The Mellin inversion yields that

$$\begin{aligned} \sum _{\begin{array}{c} (n,2)=1 \end{array}}\frac{L^{(2)}(\tfrac{1}{2}+\alpha , f \otimes \chi _{n})}{L^{(2)}(\tfrac{1}{2}+\beta , f \otimes \chi _{n})}w \left( \frac{n}{X}\right) =\frac{1}{2\pi i}\int \limits _{(2)}A\left( s,\tfrac{1}{2}+\alpha ,\tfrac{1}{2}+\beta ; f \right) X^s{\widehat{w}}(s) \textrm{d}s, \end{aligned}$$

(3.38)

where A(s, w, z; f) is defined in (3.1) and where we recall that $\widehat{w}$ is the Mellin transform of w defined by

$$\begin{aligned} \widehat{w}(s) =\int \limits ^{\infty }_0w(t)t^s\frac{\textrm{d}t}{t}. \end{aligned}$$

Now repeated integration by parts gives that for any integer $E \ge 0$,

$$\begin{aligned} {\widehat{w}}(s) \ll \frac{1}{(1+|s|)^{E}}. \end{aligned}$$

(3.39)

We evaluate the integral in (3.38) by shifting the line of integration to $\Re (s)=N(\alpha ,\beta )+\varepsilon$, where $N(\alpha ,\beta )$ is given in (1.5). Applying (3.37) and (3.39) gives that the integral on the new line can be absorbed into the O-term in (1.3). We encounter a simple pole at $s=1$ in the process whose residue is given in (3.13). This yields the main terms in (1.3) and completes the proof of Theorem 1.1.