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.

REFERENCES:

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