Explicit estimates for the zeros of Hecke L-functions

•New and improved explicit zero-free regions for Hecke L-functions.•Improved explicit zero-free region for the Dedekind zeta function.•New explicit Deuring-Heilbronn phenomenon for Hecke L-functions.•Improved explicit Deuring-Heilbronn phenomenon for the Dedekind zeta function.•Explicit zero density estimates for Hecke L-functions.

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