Experimental Mathematics

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 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 nχ only depends on the sign of the root number W(χ); indeed nχ = 0 when W (χ) = -1. Finally we give some estimates on nχ and low zeros of L(s, χ) on the critical line in terms fo the Artin conductor *#402;χ of the character χ.

Article information

Source
Experiment. Math., Volume 10, Issue 2 (2001), 237-246.

Dates
First available in Project Euclid: 30 August 2001

Permanent link to this document
https://projecteuclid.org/euclid.em/999188634

Mathematical Reviews number (MathSciNet)
MR1837673

Zentralblatt MATH identifier
1046.11083

Subjects
Primary: 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]

Keywords
Artin L-functions zeros quaternion fields

Citation

Omar, Sami. On Artin L-Functions for Octic Quaternion Fields. Experiment. Math. 10 (2001), no. 2, 237--246. https://projecteuclid.org/euclid.em/999188634


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