## Artin L-Function

An Artin -function over the Rationals encodes in a Generating Function information about how an irreducible monic Polynomial over factors when reduced modulo each Prime. For the Polynomial , the Artin -function is

where is a Legendre Symbol, which is equivalent to the Euler L-Function. The definition over arbitrary Polynomials generalizes the above expression.

Knapp, A. W. Group Representations and Harmonic Analysis, Part II.'' Not. Amer. Math. Soc. 43, 537-549, 1996.