On the critical line zeros of $L$ -- functions attached to automorphic cusp forms

TL;DR

This paper proves that a positive proportion of zeros of certain automorphic L-functions lie on the critical line, extending previous results from the full modular group to more general automorphic forms.

Contribution

It establishes the distribution of zeros on the critical line for L-functions attached to automorphic cusp forms for congruence subgroups, generalizing earlier work.

Findings

A positive proportion of zeros lie on the critical line

Extension of previous results from the full modular group

Applicable to automorphic cusp forms for $ ext{SL}_2( ext{Z})$ with level D

Abstract

We prove for L-function attached to an automorphic cusp form for the Hecke congruence group $\Gamma_0(D)$, which is also an eigenfunction of all the Hecke operators, that a positive proportion of its non-trivial zeros lie on the critical line. This result extends the work of J.L. Hafner of 1983 where the case of the full modular group is considered.