An Artin L-function over the rationals Q encodes in a generating function information about how an irreducible monic polynomial over Z factors when reduced modulo each prime. For the polynomial x^2 + 1, the Artin L-function is L(s, Q(i)/Q, sgn) = product_(p odd prime) 1/(1 - ((-1)/p) 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.

Langlands reciprocity

Emil Artin