On zeros of Hecke L-functions and their linear combinations on the critical line

Abstract

In this paper two theorems were obtained. In the first theorem it is proved that a positive proportion of non-trivial zeros lie on the critical line for L-functions attached to automorphic cusp forms for congruence-subgroups. Therefore, the class of functions satisfying a variant of Selberg’s theorem was extended. In the second theorem a new lower bound was obtained for the number of zeros of linear combinations of Hecke L-functions on the intervals of the critical line. This theorem essentially improves the previously known S.A. Gritsenko’s result of 1997.