Explicit estimates for the zeros of Hecke $L$-functions

TL;DR

This paper provides new explicit bounds and estimates for the zeros of Hecke L-functions over number fields, improving understanding of their zero-free regions, zero density, and phenomena affecting their zeros.

Contribution

It introduces improved explicit results on zeros of Hecke L-functions, with effective dependence on number field data, advancing analytic number theory techniques.

Findings

New zero-free regions for Hecke L-functions

Enhanced zero density estimates

Refined Deuring-Heilbronn phenomenon results

Abstract

Let $K$ be a number field and, for an integral ideal $\mathfrak{q}$ of $K$, let $\chi$ be a character of the narrow ray class group modulo $\mathfrak{q}$. We establish various new and improved explicit results, with effective dependence on $K$, $\mathfrak{q}$ and $\chi$, regarding the zeros of the Hecke L-function $L(s,\chi)$, such as zero-free regions, Deuring-Heilbronn phenomenon, and zero density estimates.