On the Galois structure of Artin's L-function at its critical points

This investigation evaluates a specific relation between an Artin L-function of a finite Galois extension F of Q , and its twist by an even character of degree one, taken here to mean that this character acts trivially on the action of conjugation in the Galois group of F over Q . First, an integer k is defined to be critical for a particular L-function if the Euler factors at infinity appearing in the functional equation for the given L-function are regular at k and 1 -- k. It is then shown that for such critical points k with k > 1, the twisted L-function is equal, up to an element contained in the field generated over Q by the values of the twisting character and the character appearing in the original L-function, to a product of the original L-function with a power of the Gauss sum of the twisting character, where the power of this Gauss sum corresponds to the dimension of the representation defining the character of the original L-function.