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 \(T\rightarrow \infty \).