mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, motivic ''L''-functions are a generalization of Hasse–Weil ''L''-functions to general motives over

global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function fi ...

s. The local ''L''-factor at a finite place ''v'' is similarly given by the

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...

of a

Frobenius element In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism ma ...

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 In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

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 In mathematics, the Selberg class is an axiomatic definition of a class of ''L''-functions. The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties satisfied by most functions that are co ...

. There are also conjectures concerning the values of these ''L''-functions at integers generalizing those known for the

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...

, 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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* * {{L-functions-footer Zeta and L-functions Algebraic geometry