Some explicit zero-free regions for Hecke L-functions

Let K be an algebraic number field of degree n over Q and let dK denote the absolute value of its discriminant. Let χ be a Hecke character on K with conductor F(χ). We let L(s,χ) denote the Hecke L-function associated with χ. Set Aχ=dKNK/Q(F(χ)). In this paper we present some explicit zero-free regions for Hecke L-functions. For example we prove the following results using Stechkin's device: If K≠Q, the Dedekind zeta function ζK(s) has at most one zero ρ=β+iγ with β>1−(2log⁡dK)−1 and |γ|<(2log⁡dK)−1. This zero, if it exists, has to be real and simple; If χ is a primitive Hecke character on K of order 2, then L(s,χ) has at most one zero ρ=β+iγ with β>1−(4log⁡Aχ)−1 and |γ|<(4log⁡Aχ)−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 χ on K of order |χ|≥3, L(s,χ) has no zero ρ=β+iγ in the region β>1−(15.10log⁡Aχ)−1 and |γ|<13tan⁡(π8).