: Mathematics: motivic L-function

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motivic L-function

In mathematics, motivic L-functions are a generalization of Hasse–Weil L-functions to general motives over global fields. The local L-factor at a finite place v is similarly given by the characteristic polynomial of a Frobenius element at v acting on the v-inertial invariants of the v-adic realization of the motive. For infinite places, Jean-Pierre Serre gave a recipe in (Serre 1970) for the so-called Gamma factors in terms of the Hodge realization of the motive. It is conjectured that, like other L-functions, that each motivic L-function can be analytically continued to a meromorphic function on the entire complex plane and satisfies a functional equation relating the L-function L(s, M) of a motive M to L(1 − s, M^∨), where M^∨ is the dual of the motive M.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Motivic_L-function)

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