Artin L-Function

An Artin $L$-function over the Rationals $\Bbb{Q}$ encodes in a Generating Function information about how an irreducible monic Polynomial over $\Bbb{Z}$ factors when reduced modulo each Prime. For the Polynomial $x^2+1$, the Artin $L$-function is

$$L(s,\Bbb{Q}(i)/\Bbb{Q}, \mathop{\rm sgn}\nolimits )=\prod_{p{\rm\ odd\ prime}} {1\over 1-\left({-1\over p}\right)p^{-s}},$$

where $(-1/p)$ is a Legendre Symbol, which is equivalent to the Euler L-Function. The definition over arbitrary Polynomials generalizes the above expression.

See also Langlands Reciprocity

References

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