On Artin L-functions for octic quaternion fields

We study the Artin L-function L(s, chi) associated to the unique character chi of degree 2 in quaternion fields of degree 8. We first explain how to f ind examples of quaternion octic fields with not too large a discriminant. We then develop a method yielding a quick computation of the order n(chi) o f the zero of L(s, chi) at the point s = 1/2. In all our calculations, we f ind that n(chi) only depends on the sign of the root number W(chi); indeed n(chi) = 0 when W(chi) = +1 and n(chi) = 1 when W(chi) = -1, Finally we giv e some estimates on nx and low zeros of L(s, chi) on the critical line in t erms of the Artin conductor f chi of the character chi.