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 yielding a quick computation of the order n_χ of the zero of L(s, χ) at the point s = ½. 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 and n_χ = 1 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 of the Artin conductor f_χ of the character χ.

Keywords:

• Artin L-functions
• zeros
• quaternion fields