# Motivic L-function

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(sM) of a motive M to L(1 − s, M), where M is the dual of the motive M.[1]

## Examples

Basic examples include Artin L-functions and Hasse–Weil L-functions. It is also known (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,[2] 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

1. ^ 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.
2. ^ Langlands 1980

## References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Motivic zeta function — In algebraic geometry, the motivic zeta function of a smooth algebraic variety X is the formal power series Here X(n) is the n th symmetric power of X, i.e., the quotient of Xn by the action of the symmetric group Sn, and [X(n) …   Wikipedia

• Dedekind zeta function — In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function which is obtained by specializing to the case where K is the rational numbers Q. In particular,… …   Wikipedia

• Riemann zeta function — ζ(s) in the complex plane. The color of a point s encodes the value of ζ(s): dark colors denote values close to zero and hue encodes the value s argument. The white spot at s = 1 is the pole of the zeta function; the black spots on the… …   Wikipedia

• Dirichlet L-function — In mathematics, a Dirichlet L series is a function of the form Here χ is a Dirichlet character and s a complex variable with real part greater than 1. By analytic continuation, this function can be extended to a meromorphic function on the whole… …   Wikipedia

• Дзета-функции — Эта страница информационный список. См. также основную статью: Дзета функция Римана В математике дзета функция обычно это функция родственная или аналогичная дзета функции Римана …   Википедия

• musical composition — Introduction       the act of conceiving a piece of music, the art of creating music, or the finished product. These meanings are interdependent and presume a tradition in which musical works exist as repeatable entities. In this sense,… …   Universalium

• concerto — /keuhn cher toh/; It. /kawn cherdd taw/, n., pl. concertos, concerti / tee/. Music. a composition for one or more principal instruments, with orchestral accompaniment, now usually in symphonic form. [1720 30; < It, deriv. of concertare; see… …   Universalium

• List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… …   Wikipedia

• symphony — /sim feuh nee/, n., pl. symphonies. 1. Music. a. an elaborate instrumental composition in three or more movements, similar in form to a sonata but written for an orchestra and usually of far grander proportions and more varied elements. b. an… …   Universalium

• Riemann hypothesis — The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non trivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011 …   Wikipedia