-functions are certain meromorphic functions generalizing the Riemann zeta function. They are typically defined on parts of the complex plane by a power series expressions – called the -series – which converges in that region, and then meromorphically extended to all of the complex plane by analytic continuation.
The most canonically defined class of examples of L-functions are the Artin L-functions defined for any Galois representation as the Euler products of, essentially, the characteristic polynomials of all the Frobenius homomorphisms acting via .
Most other kinds of L-functions are such as to reproduces these Artin L-functions from more “arithmetic” data:
for 1-dimensional Galois representations (hence for ) Artin reciprocity produces for each a Dirichlet character, or more generally a Hecke character? , and therefrom is built a Dirichlet L-function or Hecke L-function , respectively, which equals the corresponding Artin L-function ;
for general -dimensional Galois representations the conjecture of Langlands correspondence states that there is an automorphic representation corresponding to and an automorphic L-function built from that, which equalso the Artin L-function .
L-functions typically satisfy analogs of all the special properties enjoyed by the Riemann zeta function, such as satisfying a “functional equation” which asserts invariance under modular transformations of the parameter.
The generalized Riemann conjecture is concerned with zeros of the Dedekind zeta function for which the L-series (the Dirichlet L-function) is complicated from the classical Riemann case by the presence of the additional parameter, the Dirichlet character.
|context/function field analogy||theta function||zeta function (= Mellin transform of )||L-function (= Mellin transform of )||eta function||special values of L-functions|
|physics/2d CFT||partition function as function of complex structure of worldsheet (hence polarization of phase space) and background gauge field/source||analytically continued trace of Feynman propagator||analytically continued trace of Feynman propagator in background gauge field :||analytically continued trace of Dirac propagator in background gauge field||regularized 1-loop vacuum amplitude / regularized fermionic 1-loop vacuum amplitude / vacuum energy|
|Riemannian geometry (analysis)||zeta function of an elliptic differential operator||zeta function of an elliptic differential operator||eta function of a self-adjoint operator||functional determinant, analytic torsion|
|complex analytic geometry||section of line bundle over Jacobian variety in terms of covering coordinates on||zeta function of a Riemann surface||Selberg zeta function||Dedekind eta function|
|arithmetic geometry for a function field||Goss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)|
|arithmetic geometry for a number field||Hecke theta function, automorphic form||Dedekind zeta function (being the Artin L-function for the trivial Galois representation)||Artin L-function of a Galois representation , expressible “in coordinates” (by Artin reciprocity) as a finite-order Hecke L-function (for 1-dimensional representations) and generally (via Langlands correspondence) by an automorphic L-function (for higher dimensional reps)||class number regulator|
|arithmetic geometry for||Jacobi theta function ()/ Dirichlet theta function ( a Dirichlet character)||Riemann zeta function (being the Dirichlet L-function for Dirichlet character )||Artin L-function of a Galois representation , expressible “in coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function|
E. Kowalski, first part of Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures (pdf)
Some history is in