Counting zeros of Artin $L$-functions

Abstract

In this article, assuming Artin's (holomorphy) conjecture, we establish an explicit asymptotic formula for the number of non-trivial zeros, up to any given height $T\ge 1$, of Artin $L$-functions. As a consequence, our result yields an unconditional explicit zero-counting formula for Hecke $L$-functions over any number field. In addition, our result improves the recent work of Amberger on Dedekind and Riemann zeta functions and the previous work of Bennett-Martin-O'Bryant-Rechnitzer on Dirichlet $L$-functions for sufficiently large $T$.