Motivic L-function
In
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.[1]
Examples
Basic examples include
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
Bloch–Kato conjecture
(on special values of L-functions).
Notes
- ^ 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 w + 1 − s, where w is the weight of the motive.
- ^ Langlands 1980
References
- Zbl 0449.10022
- Scholl, Anthony (1990), "Motives for modular forms", Inventiones Mathematicae, 100 (2): 419–430, S2CID 17109327
- Serre, Jean-Pierre (1970), "Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures)", Séminaire Delange-Pisot-Poitou, 11 (2 (1969–1970) exp. 19): 1–15