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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 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 , where ''M''∨ is the ''dual'' of the motive ''M''.〔Another common normalization of the ''L''-functions consists in shifting the one used here so that the functional equation relates a value at ''s'' with one at , where ''w'' is the ''weight'' of the motive.〕
==Examples==
Basic examples include Artin ''L''-functions and Hasse–Weil ''L''-functions. It is also known , for example, that a motive can be attached to a newform (i.e. a primitive cusp form), hence their ''L''-functions are motivic.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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