In
mathematics
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, 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''.
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.
Conjectures
Several conjectures exist concerning motivic ''L''-functions. It is believed that motivic ''L''-functions should all arise as
automorphic ''L''-functions,
and hence should be part of the
Selberg class. There are also conjectures concerning the values of these ''L''-functions at integers generalizing those known for the
Riemann zeta function, such as
Deligne's conjecture on special values of ''L''-functions, the
Beilinson conjecture, and the
Bloch–Kato conjecture (on special values of ''L''-functions).
Notes
References
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Zeta and L-functions
Algebraic geometry