Artin L-Function

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.

See also

Langlands Reciprocity

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

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Artin L-Function

Cite this as:

Weisstein, Eric W. "Artin L-Function." From MathWorld--A Wolfram Web Resource.

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