Simultaneous nonvanishing of products of L-functions associated to elliptic cusp forms

Abstract

A generalized Riemann hypothesis states that all zeros of the completed Hecke L-function L* (f, s) of a normalized Hecke eigenform f on the full modular group should lie on the vertical line Re(s) = k/2. It was shown in [6] that there exists a Hecke eigenform f of weight k such that L* (f , s) not equal 0 for sufficiently large k and any point on the line segments Im(s) = t(0), k-1/2 < Re(s) < k/2 - epsilon, k/2 < Re(s) < k+1/2, for any given real number to and t(0) positive real number epsilon. This paper concerns the non-vanishing of the product L* (f , s)L* (f, w) (s, w is an element of C) on average. (C) 2020 Elsevier Inc. All rights reserved.