On computing Artin $L$-functions in the critical strip

Abstract:

This paper gives a method for computing values of certain nonabelian Artin L-functions in the complex plane. These Artin L-functions are attached to irreducible characters of degree 2 of Galois groups of certain normal extensions K of Q. These fields K are the ones for which $G = {\operatorname {Gal}}(K/{\mathbf {Q}})$ has an abelian subgroup A of index 2, whose fixed field ${\mathbf {Q}}(\sqrt d )$ is complex, and such that there is a $\sigma \in G - A$ for which $\sigma a{\sigma ^{ - 1}} = {a^{ - 1}}$ for all $a \in A$. The key property proved here is that these particular Artin L-functions are Hecke (abelian) L-functions attached to ring class characters of the imaginary quadratic field ${\mathbf {Q}}(\sqrt d )$ and, therefore, can be expressed as linear combinations of Epstein zeta functions of positive definite binary quadratic forms. Such Epstein zeta functions have rapidly convergent expansions in terms of incomplete gamma functions. In the special case $K = {\mathbf {Q}}(\sqrt { - 3} ,{a^{1/3}})$, where $a > 0$ is cube-free, the Artin L-function attached to the unique irreducible character of degree 2 of ${\operatorname {Gal}}(K/{\mathbf {Q}}) \cong {S_3}$ is the quotient of the Dedekind zeta function of the pure cubic field $L = {\mathbf {Q}}({a^{1/3}})$ by the Riemann zeta function. For functions of this latter form, representations as linear combinations of Epstein zeta functions were worked out by Dedekind in 1879. For $a = 2,3,6$ and 12, such representations are used to show that all of the zeroes $\rho = \sigma + it$ of these L-functions with $0 < \sigma < 1$ and $|t| \leqslant 15$ are simple and lie on the critical line $\sigma = 1/2$. These methods currently cannot be used to compute values of L-functions with $\operatorname {Im} (s)$ much larger than 15, but approaches to overcome these deficiencies are discussed in the final section.