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 {{harv|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 {{nowrap|L(1 − s, M∨)}}, 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 {{nowrap|w + 1 − s}}, where w is the weight of the motive.
Examples
Basic examples include Artin L-functions and Hasse–Weil L-functions. It is also known {{harv|Scholl|1990}}, 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,{{harvnb|Langlands|1980}} 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
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References
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| publisher=AMS
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| mr=0546622 | zbl=0449.10022
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| contribution=L-functions and automorphic representations
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| s2cid=17109327
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| author-link=Jean-Pierre Serre
| title=Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures)
| journal=Séminaire Delange-Pisot-Poitou
| year=1970
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}}
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