Explicit zero-free regions for automorphic $L$-functions

Abstract

Let $L(s,f)$ be the $L$-function associated with a newform $f$ of even weight $k$, squarefree level $N$ and trivial nebentypus. In this paper, we establish a new explicit zero-free region for $L(s,f)$. More precisely, we prove that $L(s,f)$ does not vanish in the region $\Re(s)\geq 1-\frac{1}{C\log(kN\max(1,|\Im(s)|))}$ with $C=16.7053$ if $|\Im(s)|\geq 1$ or $|\Im(s)|\leq \frac{0.30992}{\log(kN)}$ and $C=16.9309$ if $\frac{0.30992}{\log(kN)}<|\Im(s)|\leq 1$. This improves a result of Hoey et al. where $445.994$ was shown to be an admissible value for $C$.