Some explicit zero-free regions for Hecke L-functions

Abstract

Let K be an algebraic number field of degree n over Q and let d(K) denote the absolute value of its discriminant. Let chi be a Hecke character on K with conductor F(chi). We let L(s,chi) denote the Hecke L-function associated with chi. Set A(chi) = d(K)N(K/Q)(F(chi)). In this paper we present some explicit zero-free regions for Hecke 1,L-functions. For example we prove the following results using Stechkin's device: If K not equal Q, the Dedekind zeta function zeta(K) (s) has at most one zero rho = beta + i gamma with beta > 1 - (2 log d(K))(-1) and vertical bar gamma vertical bar < (2 log d(K))(-1). This zero, if it exists, has to be real and simple; If x is a primitive Hecke character on K of order 2, then L(s,x) has at most one zero rho = beta + i gamma with beta > 1 - (4 log A(chi))(-1) and vertical bar gamma vertical bar < (4 log A chi)(-1). If such a zero exists, it has to be real and simple. Moreover, using approaches due to Heath-Brown and to Kadiri, we show that for a primitive Hecke character chi on K of order vertical bar chi vertical bar >= 3, L(s, chi) has no zero rho = beta + i gamma in the region beta > 1 - (15.10 log A chi)(-1) and vertical bar gamma vertical bar < 1/3 tan(pi/8). (C) 2014 Elsevier Inc. All rights reserved.