$L$-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 $L$-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 $\sigma \colon Gal \longrightarrow GL_n(\mathbb{C})$ as the Euler products of, essentially, the characteristic polynomials of all the Frobenius homomorphisms acting via $\sigma$.
Most other kinds of L-functions are such as to reproduces these Artin L-functions from more “arithmetic” data:
for 1-dimensional Galois representations $\sigma$ (hence for $n = 1$) Artin reciprocity produces for each $\sigma$ a Dirichlet character, or more generally a Hecke character? $\chi$, and therefrom is built a Dirichlet L-function or Hecke L-function $L_\chi$, respectively, which equals the corresponding Artin L-function $L_\sigma$;
for general $n$-dimensional Galois representations $\sigma$ the conjecture of Langlands correspondence states that there is an automorphic representation $\pi$ corresponding to $\sigma$ and an automorphic L-function $L_\pi$ built from that, which equalso the Artin L-function $L_\sigma$.
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 $\theta$ | zeta function $\zeta$ (= Mellin transform of $\theta(0,-)$) | L-function $L_{\mathbf{z}}$ (= Mellin transform of $\theta(\mathbf{z},-)$) | eta function $\eta$ | special values of L-functions |
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physics/2d CFT | partition function $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of complex structure $\mathbf{\tau}$ of worldsheet $\Sigma$ (hence polarization of phase space) and background gauge field/source $\mathbf{z}$ | analytically continued trace of Feynman propagator $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$ | analytically continued trace of Feynman propagator in background gauge field $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$ | analytically continued trace of Dirac propagator in background gauge field $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s$ | regularized 1-loop vacuum amplitude $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / regularized fermionic 1-loop vacuum amplitude $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / vacuum energy $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$ |
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 $\theta(\mathbf{z},\mathbf{\tau})$ of line bundle over Jacobian variety $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$ | 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 $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the trivial Galois representation) | Artin L-function $L_{\mathbf{z}}$ of a Galois representation $\mathbf{z}$, 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 $\cdot$ regulator | |
arithmetic geometry for $\mathbb{Q}$ | Jacobi theta function ($\mathbf{z} = 0$)/ Dirichlet theta function ($\mathbf{z} = \chi$ a Dirichlet character) | Riemann zeta function (being the Dirichlet L-function $L_{\mathbf{z}}$ for Dirichlet character $\mathbf{z} = 0$) | Artin L-function of a Galois representation $\mathbf{z}$ , 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 |
Stephen Gelbart, section C starting on p. 14 (190) of An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219 (web)
E. Kowalski, first part of Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures (pdf)
Dorian Goldfeld, Joseph Hundley, chapter 2 of Automorphic representations and L-functions for the general linear group, Cambridge Studies in Advanced Mathematics 129, 2011 (pdf)
wikipedia L-function, Dirichlet L-function, special values of L-function, generalized Riemann hypothesis, Artin L-function, equivariant L-function, functional equation), modularity theorem
Terrence Tao’s blog: Distinguished Lecture Series III: Shou-wu Zhang, “Triple L-series and effective Mordell conjecture”