ON ARTIN L-FUNCTIONS FOR OCTIC QUATERNION FIELDS

SAMI OMAR

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 = \frac{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 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.