On Artin L-Functions for Octic Quaternion Fields

Abstract

We study the Artin L-function L(s, χ) associated to the unique character χ of degree 2 in quaternion fields of degree 8. We first explain how to find examples of quaternion octic fields with not too large a discriminant. We then develop a method using a quick compuation of the order n_*#967; of the zero of L(s, χ) at the point s = 1/2. In all our calculations, we find that n_χ only depends on the sign of the root number W(χ); indeed n_χ = 0 when W (χ) = -1. Finally we give some estimates on n_χ and low zeros of L(s, χ) on the critical line in terms fo the Artin conductor *#402;_χ of the character χ.