Zeros of a family of approximations of Hecke L-functions associated with cusp forms

Abstract

We consider a family of approximations of a Hecke L-function $L_f(s)$ attached to a holomorphic cusp form f of positive integral weight k with respect to the full modular group. These families are of the form

$$\begin{aligned} L_f(X;s):=\sum _{n\le X}\frac{a(n)}{n^s}+(-1)^{k/2}(2\pi )^{-(1-2s)}\frac{\Gamma \left( \tfrac{k+1}{2}-s\right) }{\Gamma \left( \tfrac{k-1}{2}+s\right) }\sum _{n\le X}\frac{a(n)}{n^{1-s}}, \end{aligned}$$

where $s=\sigma +it$ is a complex variable and a(n) is a normalized Fourier coefficient of f. From an approximate functional equation, one sees that $L_f(X;s)$ is a good approximation to $L_f(s)$ when $X=t/2\pi$. We obtain vertical strips where most of the zeros of $L_f(X;s)$ lie. We study the distribution of zeros of $L_f(X;s)$ when X is independent of t. For $X=1$ and 2, we prove that all the complex zeros of $L_f(X;s)$ lie on the critical line $\sigma =1/2$. We also show that as $T\rightarrow \infty$ and $X=T^{o(1)}$, $100\,\%$ of the complex zeros of $L_f(X;s)$ up to height T lie on the critical line. Here by $100\,\%$ we mean that the ratio between the number of simple zeros on the critical line and the total number of zeros up to height T approaches 1 as $T\rightarrow \infty$.