1 Background

The first named author had shown in [4] that given any real number \(t_0\) and \(\epsilon >0\), then for k large enough, the average of the normalized L-functions \(L^*(f,s)\) with f varying over a basis of Hecke eigenforms of weight k on \(SL_2(\mathbb {Z})\) does not vanish on the line segment \(Im s=t_0\), \((k-1)/2< Re s<k/2-\epsilon , k/2+\epsilon<Re s<(k+1)/2\). In [5], the authors extend the result for cuspidal Hecke eigenforms of half integer weight. In what follows, we prove a non-vanishing result for sums of L-functions associated to cuspidal Hecke eigenforms of half-integral weight \(k+1/2\) where \(k\in 2\mathbb {Z}\) on level 4 in the plus space. For this, we determine the Fourier coefficients of the projected kernel function.

2 Projection onto the plus space

If \(z\in \mathbb {C}-\{0\}\) and \(x\in \mathbb {C}\), let \(z^x=e^{x\log z}\) where \(-\pi <arg z\le \pi \). Let \(q=e^{2\pi iz}\). The symbol \(\left( \frac{c}{d}\right) \) for c, \(d\ne 0\) integers is given in [6]. In addition, for more details about modular forms of half-integer weight, the reader can refer to [6]. We introduce the group \(\mathcal {C}\) consisting of all pairs \((A,\phi (z))\) where \(A=\left( \begin{array}{l@{\quad }l} a &{} b \\ c &{} d \\ \end{array} \right) \in GL_2^+(\mathbb {R})\) and \(\phi :\mathbb {H}\rightarrow \mathbb {C}\) is a holomorphic function with \(|\phi (z)|=(det A)^{-1/4}|cz+d|^{1/2}\) satisfying the following group law \((A,\phi (z))(B,\psi (z))=(AB,\phi (Bz)\psi (z))\). Throughout if \(\zeta =(A,\phi (z))\), k integer and \(f:\mathbb {H}\rightarrow \mathbb {C}\) is a function, we put \(f|_{k+1/2}\zeta =f|\zeta =\phi (z)^{-2k-1}f(Az)\). We have an embedding \(\Gamma _0(4)\hookrightarrow \mathcal {C}\) given by \(A\rightarrow A^*:=(A,j(A,z))\) where \(j(A,z)=\left( \frac{c}{d}\right) \left( \frac{-4}{d}\right) ^{-1/2}(cz+d)^{1/2}\) if \(A=\left( \begin{array}{l@{\quad }l} a &{} b \\ c &{} d \\ \end{array} \right) \). Recall also that \(M_{k+1/2}(4)\) consists of complex valued functions f holomorphic on \(\mathbb {H}\) which satisfy \(f|A^*=f\) for all \(A\in \Gamma _0(4)\) and which are holomorphic at the cusps. Also, \(S_{k+1/2}(4)\) is the subspace of \(M_{k+1/2}(4)\) consisting of f which vanish at the cusps. We also define \(M^+_{k+1/2}(4)\) to be the subspace of \(M_{k+1/2}(4)\) consisting of functions whose nth Fourier coefficients vanish whenever \((-1)^kn\equiv 2,3\mod (4)\). We also put \(S^+_{k+1/2}(4)=S_{k+1/2}(4)\cap M^+_{k+1/2}(4)\). Let L(f, s) be the L-function associated to cusp forms \(f\in S^+_{k+1/2}(4)\) defined by \(L(f,s)=\sum _{(-1)^kn\equiv 0,1 \ (4)}a_f(n)n^{-s}\) for \(\sigma :=Re \ s \gg 1\) where \(a_f(n)\) is the nth Fourier coefficient of f. The completed L-function is defined as \(L^*(f,s):=(2\pi )^{-s}2^s\Gamma (s)L(f,s)\).

The kernel function of the map \(f\rightarrow L^*(f,s)\)in the case of half integral weight on \(\Gamma _0(4)\) is given in [5] by

$$\begin{aligned} R_{s,k}(z)&=\gamma _k(s)\sum _{A}\left( \frac{c}{d}\right) \left( \frac{-4}{d}\right) ^{k+1/2} (cz+d)^{-(k+1/2)}\left( \frac{az+b}{cz+d}\right) ^{-s}; \nonumber \\&\qquad \quad 1<\sigma <k-1/2 \end{aligned}$$
(1)

where

$$\begin{aligned} \gamma _k(s)=\frac{1}{2} e^{\pi i s/2}\Gamma (s)\Gamma (k+1/2-s), \end{aligned}$$
(2)

and the sum runs over all matrices \(A=\left( \begin{array}{cc} a &{}\quad b \\ c &{}\quad d \\ \end{array} \right) \) in \(\Gamma _0(4)\).

What we will do is to determine non-vanishing of L-functions for modular forms of half integral weight in the plus space. In order to do so, one needs to define the kernel function that acts on that space.

For the sake of efficiency, we will write

$$\begin{aligned} R_{s,k}(z)=\gamma _k(s)\sum _{A} z^{-s}|_{k+1/2}A^* , \end{aligned}$$
(3)

where \(A \in \Gamma _0(4)\). In order to determine the kernel function for modular forms in the plus space, one has to use the projection operator pr as given in [2]. For \(g \in S_k(\Gamma _0(4))\), we have

$$\begin{aligned} g|pr=(-1)^{(k+1)/2}\frac{1}{3\sqrt{2}}\left( \sum _{\nu \mod 4}g|\zeta A^*_{\nu }\right) +\frac{1}{3}g, \end{aligned}$$
(4)

where

$$\begin{aligned} \zeta =\left( \left( \begin{array}{cc} 4 &{}\quad 1 \\ 0 &{}\quad 4 \\ \end{array} \right) ,e^{(2k+1)\pi i/4}\right) , \ \ \ \ \ \ A^*_{\nu }=\left( \left( \begin{array}{cc} 1 &{}\quad 0 \\ 4\nu &{}\quad 1 \\ \end{array} \right) , (4\nu z+1)^{-(k+1/2)}\right) . \end{aligned}$$

The Fourier expansion of the projected function g|pr (in the more general case of \(\Gamma _0(4N)\)) is given in ([2] Proposition 3) in terms of the Fourier coefficients of g at different cusps. In particular on \(\Gamma _0(4)\), we get the following lemma for \(R_{s,k}|pr\).

Lemma 1

$$\begin{aligned} R_{s,k}|pr&= \frac{2}{3} \sum _{n\ge 1, n\equiv 0(4)}\left( a_s(n)+(1-(-1)^ki)2^{2k-1}i^{n/4}a_s^{1}(n/4)\right) q^n\nonumber \\&\quad +\frac{2}{3}\sum _{n\ge 1, (-1)^kn\equiv 1(4)}\left( a_s(n)+2^{k-1}\left( \frac{(-1)^kn}{2}\right) a_s^{1/2}(n)\right) q^n, \end{aligned}$$
(5)

where

$$\begin{aligned} a_s(n)&=(2\pi )^s\Gamma (k+1/2-s)n^{s-1}+\frac{1}{2}(2\pi i)^{k+1/2}n^{k-1/2}\frac{\Gamma (s)\Gamma (k+1/2-s)}{\Gamma (k+1/2)}\\&\quad \times \sum _{\begin{array}{c} ac>0\\ (a,c)=1, 4|c \end{array}}\left( \frac{c}{a}\right) \left( \frac{-4}{a}\right) ^{k+1/2}c^{-(k+1/2)}\left( \frac{c}{a}\right) ^s\\&\qquad \left( e^{2\pi ina'/c}e^{\pi is/2} {_1}F_1(s,k+1/2;-2\pi in/ac) \right. \\&\left. \qquad \quad + \,e^{-2\pi ina'/c}e^{-\pi is/2} {_1}F_1(s,k+1/2;2\pi in/ac)\right) , \end{aligned}$$
$$\begin{aligned} a_s^1(n)&=2\times 4^s(2\pi )^s\Gamma (k+1/2-s)n^{s-1} \\&\quad +4^s(2\pi i)^{k+1/2}n^{k-1/2}\frac{\Gamma (s)\Gamma (k+1/2-s)}{\Gamma (k+1/2)}\\&\quad \times \sum _{\begin{array}{c} c\ge 1, c\equiv 1\mod 2\\ d(c)*, ad-bc=1\\ d\equiv -c \mod 4 \end{array}}\left( \frac{4c}{d}\right) \left( \frac{-4}{-c}\right) ^{k+1/2}c^{-(k+1/2)}\left( \frac{c}{a}\right) ^s\\&\qquad \left( e^{2\pi ind/c}e^{\pi is/2} {_1}F_1(s,k+1/2;-2\pi in/ac) \right. \\&\left. \qquad \quad +\, \,e^{-2\pi ind/c}e^{-\pi is/2} {_1}F_1(s,k+1/2;2\pi in/ac)\right) \end{aligned}$$

and

$$\begin{aligned} a_s^{1/2}(n)&=2\times 4^s(2\pi )^s\Gamma (k+1/2-s)n^{s-1} \\&\quad +4^s(2\pi i)^{k+1/2}n^{k-1/2}\frac{\Gamma (s)\Gamma (k+1/2-s)}{\Gamma (k+1/2)}\\&\quad \times \sum _{\begin{array}{c} c\ge 1, c\equiv 1\mod 2\\ d(c)*, ad-2bc=1\\ d\equiv -c \mod 4 \end{array}}\left( \frac{4c}{d}\right) \left( \frac{-4}{-c}\right) ^{k+1/2}c^{-(k+1/2)}\left( \frac{c}{a}\right) ^s\\&\qquad \left( e^{2\pi ind/c}e^{\pi is/2} {_1}F_1(s,k+1/2;-2\pi in/ac) \right. \\&\left. \qquad \quad +\, \,e^{-2\pi ind/c}e^{-\pi is/2} {_1}F_1(s,k+1/2;2\pi in/ac)\right) . \end{aligned}$$

Proof

As for the expansion at the infinite cusp, we follow similar steps as in [4] with minor changes due to the fact of considering real weight. We consider first the case \(ac=0\) where we get the elements \(\left( \begin{array}{ll} 1 &{}\quad n \\ 0 &{}\quad 1 \\ \end{array} \right) \) in \(\Gamma _0(4)\). The contribution of the terms with \(ac=0\) to the sum thus is given by

$$\begin{aligned} (2\pi )^s\Gamma (k+1/2-s)\sum _{n\ge 1}n^{s-1}e^{2\pi inz}. \end{aligned}$$
(6)

To determine the contribution of the terms with \(ac\ne 0\), we consider

$$\begin{aligned}&\int _{iC}^{iC+1}\sum _{\left( \begin{array}{cc} a &{} b \\ c &{} d \\ \end{array} \right) \in \Gamma _0(4); ac\ne 0}\left( \frac{c}{d}\right) \left( \frac{-4}{d}\right) ^{k+1/2}(cz+d)^{-(k+1/2)} \left( \frac{az+b}{cz+d}\right) ^{-s}e^{-2\pi inz}dz, \nonumber \\&\quad C>0. \end{aligned}$$
(7)

We get

$$\begin{aligned} R_{s,k}(z)=\sum _{n\ge 1}a_{s}(n)e^{2\pi inz} \end{aligned}$$
(8)

where the Fourier coefficients \(a_{s}(n)\) are given by

$$\begin{aligned} a_s(n)&=(2\pi )^s\Gamma (k+1/2-s)n^{s-1}+\frac{1}{2}(2\pi i)^{k+1/2}n^{k-1/2}\frac{\Gamma (s)\Gamma (k+1/2-s)}{\Gamma (k+1/2)}\\&\quad \times \sum _{\begin{array}{c} ac>0\\ (a,c)=1, 4|c \end{array}}\left( \frac{c}{a}\right) \left( \frac{-4}{a}\right) ^{k+1/2}c^{-(k+1/2)}\left( \frac{c}{a}\right) ^s\\&\qquad \left( e^{2\pi ina'/c}e^{\pi is/2} {_1}F_1(s,k+1/2;-2\pi in/ac) \right. \\&\left. \qquad \quad +\, e^{-2\pi ina'/c}e^{-\pi is/2} {_1}F_1(s,k+1/2;2\pi in/ac)\right) . \end{aligned}$$

Here \(a'\) is an inverse of a modulo c, and \({_1}F_1(\alpha , \beta ;z)\) is Kummer’s degenerate hypergeometric function.

As for the expansion at the cusp 1, we have to determine the expansion of

$$\begin{aligned} R_{s,k}|\eta ^{1}(z), \end{aligned}$$
(9)

where \(\eta ^{1}(z)=\left( \left( \begin{array}{ll} 1 &{}\quad 0 \\ 1 &{}\quad 1 \\ \end{array} \right) , (z+1)^{k+1/2}\right) \).

We have to determine \(a_s^1(n)\) such that

$$\begin{aligned} R_{s,k}|\eta ^{1}(z)=(z+1)^{-(k+1/2)}R_{s,k}\left( \frac{z}{z+1}\right) =\sum _{n\ge 1}a_s^1(n)e^{2\pi inz/4} \end{aligned}$$
(10)

We have

$$\begin{aligned} R_{s,k}|\eta ^{1}(z)&= \gamma _{k}(s)(z+1)^{-(k+1/2)}\sum _{n\in \mathbb {Z}}\left( \frac{z}{z+1}+n\right) ^{-s} \nonumber \\&\quad +2\gamma _{k}(s)\sum _{\begin{array}{c} c>0,d\\ (d,c)=1, ad-bc=1 \end{array}}\left( \frac{4}{d}\right) \left( \frac{c+d}{d}\right) \left( \frac{-4}{d}\right) ^{k+1/2}\nonumber \\&\quad \times ((c+d)z+d)^{-(k+1/2)}\left( \frac{(a+b)z+b}{(c+d)z+d}\right) ^{-s} \end{aligned}$$
(11)

where a and b chosen such that \(ad-bc=1\).

Replace \(c+d\) by c and \(a+b\) by a to get

$$\begin{aligned} R_{s,k}|\eta ^{1}(z)&= \gamma _{k}(s)(z+1)^{-(k+1/2)}\sum _{n\in \mathbb {Z}}\left( \frac{z(n+1)+n}{z+1}\right) ^{-s} \nonumber \\&\quad +2\gamma _{k}(s)\sum _{\begin{array}{c} c>0,d\\ (d,c)=1, ad-bc=1\\ c\equiv d \ mod \ 4 \end{array}}\left( \frac{4c}{d}\right) \left( \frac{-4}{d}\right) ^{k+1/2} \nonumber \\&\quad \times (cz+d)^{-(k+1/2)}\left( \frac{az+b}{cz+d}\right) ^{-s}\nonumber \\&= 2\gamma _{k}(s)\sum _{\begin{array}{c} c>0,d\\ (d,c)=1, ad-bc=1\\ d\equiv -c \ mod \ 4, c\equiv 1\ mod \ 2 \end{array}}\left( \frac{4c}{d}\right) \left( \frac{-4}{d}\right) ^{k+1/2} \nonumber \\&\quad \times (cz+d)^{-(k+1/2)}\left( \frac{az+b}{cz+d}\right) ^{-s}. \end{aligned}$$
(12)

Thus we get

$$\begin{aligned} R_{s,k}|\eta ^{1}(z)&=2\times 4^s\gamma _{k}(s)\sum _{\begin{array}{c} c\ge 1, c\equiv 1\ mod \ 2 \end{array}}\left( \frac{-4}{-c}\right) ^{k+1/2}\sum _{\begin{array}{c} d(c)^*, ad-bc=1\\ d\equiv -c \ mod \ 4 \end{array}}\left( \frac{4c}{d}\right) \nonumber \\&\quad . \sum _{r\in \mathbb {Z}}\left( 4c(z/4+r)+d\right) ^{-k-1/2}\left( \frac{a(z/4+r)+b}{c(z/4+r)+d}\right) ^{-s}, \end{aligned}$$
(13)

where \(d(c)^*\) means that d runs through a primitive residue system modulo c. We proceed with finding \(a_s^1(n)\) similarly as we determined the expansion at infinity to get

$$\begin{aligned} a_s^1(n)&=2\times 4^s(2\pi )^s\Gamma (k+1/2-s)n^{s-1} \\&\quad +4^s(2\pi i)^{k+1/2}n^{k-1/2}\frac{\Gamma (s)\Gamma (k+1/2-s)}{\Gamma (k+1/2)}\\&\quad \times \sum _{\begin{array}{c} c\ge 1, c\equiv 1\ mod \ 2\\ d(c)^*, ad-bc=1\\ d\equiv -c \ mod \ 4 \end{array}}\left( \frac{4c}{d}\right) \left( \frac{-4}{-c}\right) ^{k+1/2}c^{-(k+1/2)}\left( \frac{c}{a}\right) ^s\\&\qquad \left( e^{2\pi ind/c}e^{\pi is/2} {_1}F_1(s,k+1/2;-2\pi in/ac) \right. \\&\left. \qquad \quad + e^{-2\pi ind/c}e^{-\pi is/2} {_1}F_1(s,k+1/2;2\pi in/ac)\right) . \end{aligned}$$

What remains is to determine the expansion at the cusp 1/2. We have

$$\begin{aligned} R_{s,k}|\eta ^{1/2}(z), \end{aligned}$$
(14)

where \(\eta ^{1/2}(z)=\left( \left( \begin{array}{ll} 1 &{}\quad 0 \\ 2 &{}\quad 1 \\ \end{array} \right) , (2z+1)^{k+1/2}\right) \).

We have to determine \(a_s^{1/2}(n)\) such that

$$\begin{aligned} R_{s,k}|\eta ^{1/2}(z)=(2z+1)^{-(k+1/2)}R_{s,k}\left( \frac{z}{2z+1}\right) =\sum _{n\ge 1, n\equiv 1 \ mod \ 4}a_s^1(n)e^{2\pi inz/4}. \end{aligned}$$
(15)

We argue in the same way as at the cusp 1 and we obtain

$$\begin{aligned} a_s^{1/2}(n)&=2\times 4^s(2\pi )^s\Gamma (k+1/2-s)n^{s-1}\\&\quad +4^s(2\pi i)^{k+1/2}n^{k-1/2}\frac{\Gamma (s)\Gamma (k+1/2-s)}{\Gamma (k+1/2)}\\&\quad \times \sum _{\begin{array}{c} c\ge 1, c\equiv 1\mod 2\\ d(c)^*, ad-2bc=1\\ d\equiv -c \ mod \ 4 \end{array}}\left( \frac{4c}{d}\right) \left( \frac{-4}{-c}\right) ^{k+1/2}c^{-(k+1/2)}\left( \frac{c}{a}\right) ^s\\&\qquad \left( e^{2\pi ind/c}e^{\pi is/2} {_1}F_1(s,k+1/2;-2\pi in/ac) \right. \\&\left. \qquad \quad + e^{-2\pi ind/c}e^{-\pi is/2} {_1}F_1(s,k+1/2;2\pi in/ac)\right) . \end{aligned}$$

\(\square \)

3 Statement of results

Throughout this section, we will assume that \(k\in 2\mathbb {Z}\). If \(f=\sum _{n\equiv 0,1 (4)}a_f(n)e^{2\pi inz}\in S^+_{k+1/2}(4)\), the L-function associated to f is defined by

$$\begin{aligned} L(f,s)=\sum _{n\equiv 0,1(4)}\frac{a_f(n)}{n^s}, \ \ \ \ \ \ \ \ \ \ \ \sigma \gg 1. \end{aligned}$$

The completed L-function defined as \(L^*(f,s)\) has the following functional equation

$$\begin{aligned} L^*(f|W_4,k+1/2-s)=L^*(f,s) \end{aligned}$$
(16)

where \(W_4\) is the Fricke involution on \(S_{k+1/2}(4)\) defined by

$$\begin{aligned} f|W_4(z)=(-2\pi iz)^{-k-1/2}f(-1/4z), \end{aligned}$$
(17)

cf [6]. Using [1], we know that \(S^+_{k+1/2}(4)\) has an orthogonal basis of Hecke eigenforms with respect to all Hecke operators \(T^+(p^2)\), p prime. Let \(\{f_{k,1},f_{k,2},\ldots f_{k,d}\}\) be such a basis of Hecke eigenforms, where d is the dimension of the space \(S^+_{k+1/2}(4)\). We know as well that we can choose the coefficients a(n) of the eigenforms to be real.

Since our projection operator is hermitian, we have for \(f\in S^+_{k+1/2}(4)\),

$$\begin{aligned} \langle f,R_{\bar{s},k}\rangle =\langle f|pr,R_{\bar{s},k}\rangle =\langle f,R_{\bar{s},k}|pr\rangle =c_kL^*(f,k+1/2-s) \end{aligned}$$
(18)

where \(c_k\) is a non-zero constant depending only on k. We therefore deduce that

$$\begin{aligned} \langle f,R_{\bar{s},k}|pr\rangle =c_kL^*(f,k+1/2-s) \end{aligned}$$
(19)

for all \(f\in S^+_{k+1/2}(4).\) As a result, we have

$$\begin{aligned} R_{s,k}|pr=c^{'}_k\sum _{j=1}^d\frac{1}{\langle f_{k,j}, f_{k,j}\rangle }L^*(f_{k,j},k+1/2-s)f_{k,j} \end{aligned}$$
(20)

(where \(c^{'}_k=\) constant) and we get

$$\begin{aligned} R_{s,k}|pr=c^{'}_k\sum _{j=1}^d\frac{1}{\langle f_{k,j}, f_{k,j}\rangle }L^*(f_{k,j}|W_4,s)f_{k,j}. \end{aligned}$$
(21)

Theorem 1

Let \(t_0\in \mathbb {R}\). Then there exists a constant \(C(t_0,\epsilon )>0\) depending only on \(t_0\) and \(\epsilon \) such that for \(k>C(t_0,\epsilon )\) and \(k\in 2\mathbb {Z}\), the function

$$\begin{aligned} \sum _{j=1}^{d}\frac{1}{\langle f_{k,j}, f_{k,j}\rangle }L^*(f_{k,j}|W_4,s) \end{aligned}$$
(22)

does not vanish at any point \(s=\sigma +it_0\) with \(t=t_0, k/2-1/4<\sigma <k/2+3/4\).

Proof

Assume that \(\sum _{j=1}^{d}\frac{1}{\langle f_{k,j}, f_{k,j}\rangle }L^*(f_{k,j}|W_4,s)=0\), for s as given in the theorem. This implies in particular that the first Fourier coefficient of \(R_{s,k}|pr\) is 0. Thus dividing by \(\frac{2}{3}4^s(2\pi )^s\Gamma (k+1/2-s)\), we have for \(k\in 2\mathbb {Z}\),

$$\begin{aligned} 4^{-s}&+\frac{(2\pi i)^{k+1/2-s} }{2\times 4^s\Gamma (k+1/2-s)}\sum _{\begin{array}{c} ac>0\\ (a,c)=1, 4|c \end{array}}\left( \frac{c}{a}\right) \left( \frac{-4}{a}\right) ^{k+1/2}c^{-(k+1/2)}\left( c/a\right) ^s\nonumber \\&\left( e^{2\pi ia'/c}e^{\pi is/2} {_1}f_1(s,k+1/2;-2\pi i/ac) \right. \nonumber \\&\left. \quad + e^{-2\pi ia'/c}e^{-\pi is/2} {_1}f_1(s,k+1/2;2\pi i/ac)\right) \nonumber \\&+ 2 +\frac{(2\pi i)^{k+1/2-s}}{\Gamma (k+1/2-s)}\sum _{\begin{array}{c} c\ge 1, c\equiv 1\ mod \ 2\\ d(c)^*, ad-2bc=1\\ d\equiv -c \ mod \ 4 \end{array}}\left( \frac{4c}{d}\right) \left( \frac{-4}{-c}\right) ^{k+1/2}c^{-(k+1/2)}\left( c/a\right) ^s\nonumber \\&\left( e^{2\pi id/c}e^{\pi is/2} {_1}f_1(s,k+1/2;-2\pi i/ac) \right. \nonumber \\&\left. \quad + e^{-2\pi id/c}e^{-\pi is/2} {_1}f_1(s,k+1/2;2\pi i/ac)\right) =0, \end{aligned}$$
(23)

where

$$\begin{aligned} {_1}f_1(\alpha ,\beta ;z)=\frac{\Gamma (\alpha )\Gamma (\beta -\alpha )}{\Gamma (\beta )}{_1}F_1(\alpha ,\beta ;z). \end{aligned}$$
(24)

Let \(s=k/2+1/4-\delta +it_0\), where \(0\le \delta \le 1/2\). We have in [4] that

$$\begin{aligned} |{_1}f_1(s,k+1/2;2\pi i/ac)|\le 1. \end{aligned}$$
(25)

As a result, we have

$$\begin{aligned}&2-4^{-k/2-1/4+\delta }\le |2+4^{-s}|\nonumber \\&\quad \le \frac{(2\pi )^{k/2+1/4+\delta }}{2\times 4^{k/2+1/4-\delta }|\Gamma (\frac{k}{2}+\frac{1}{4}+\delta -it_0)|} \sum _{\begin{array}{c} ac>0\\ (a,c)=1, 4|c \end{array}}\frac{1}{c^{k+1/2}}(e^{-\pi t_0/2}+e^{\pi t_0/2})\nonumber \\&\qquad + \frac{(2\pi )^{k/2+1/4+\delta }}{|\Gamma (\frac{k}{2}+\frac{1}{4}+\delta -it_0)|} \sum _{\begin{array}{c} c\ge 1, c\equiv 1\ mod \ 2\\ d(c)^*, ad-2bc=1\\ d\equiv -c \ mod \ 4 \end{array}}\frac{1}{c^{k+1/2}}(e^{-\pi t_0/2}+e^{\pi t_0/2}) \end{aligned}$$
(26)

We now take the limit of both sides as \(k\rightarrow \infty \) to get \(2\le 0\) and hence a contradiction.

\(\square \)

It is important to note that the above result translates into the same non-vanishing result for \(L^*(f_{k,j},s)\) and this due to the fact that the strip we are considering is invariant under \(s\rightarrow k+1/2-s\).

Corollary 1

Let \(s_0=\sigma +it_0\) with \( k/2-1/4<\sigma <k/2+3/4\), then for \(k>C(t_0,\epsilon )\), k even, there exists a Hecke eigenform \(f\in S^+_{k+1/2}(4)\) whose L-values do not vanish at \(s_0\).