An explicit zero-free region for the Dirichlet L-functions

Abstract

Let $L(s,\chi)$ be the Dirichlet $L$-function associated to a non-principal primitive character $\chi$ modulo $q$ with $3\le q \le 400\,000$. We prove a new explicit zero-free region for $L(s,\chi)$: $L(s,\chi)$ does not vanish in the region $\Re s \ge 1-\frac1{R \log \left(q\max(1,| \Im s|)\right) }$ with $R=5.60$. This improves a result of McCurley where $9.65$ was shown to be an admissible value for $R$.