## Experimental Mathematics

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

### 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.

#### 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