View source for L-function

{{MSC|11Mxx}}
{{TEX|done}}

A generalization of the
[[Zeta-function|zeta-function]] at the cost of introducing characters (cf.
[[Character of a group|Character of a group]]). The $L$-functions form a complicated class of special functions of a complex variable, defined by a
[[Dirichlet series|Dirichlet series]] or an
[[Euler product|Euler product]] with characters. They are the basic instrument for studying by analytic methods the arithmetic of corresponding mathematical objects: the field of rational numbers, algebraic fields, algebraic varieties over finite fields, etc. The simplest representatives of $L$-functions are the Dirichlet $L$-functions (cf.
[[Dirichlet-L-function|Dirichlet $L$-function]]). The remaining $L$-functions are more or less close analogues and generalizations of these $L$-functions.



====Comments====
Nowadays $L$-functions comprise a very large class of functions which are attached to representations of the
[[Galois group|Galois group]] $\def\Gal{\textrm{Gal}}\Gal(\overline{\Q}/\Q)$. For example, choose a representation $\rho : G \to {\textrm{GL}}(n,\C)$ of the Galois group $G$ of an algebraic number field $K$ (cf.
[[Representation of a group|Representation of a group]]). For each prime $p$, let $F_p$ be a Frobenius element in $G$. Then the function

$$L(p,s) = \prod_p \det(\textrm{Id} - \rho(F_p) p^{-s})^{-1}$$
is the Artin $L$-series corresponding to $\rho$. In a similar way, the action of $\Gal(\overline{\Q}/\Q)$ on the $l^n$-torsion points of an elliptic curve $E$, defined over $\Q$, gives rise to the Hasse–Weil $L$-function of $E$. There exists a large body of fascinating conjectures about these $L$-functions, which, on the one hand, relate them to automorphic forms (Langlands' conjectures) and, on the other hand, relate values at integral points to algebraic-geometric invariants (Beilinson's conjectures).

====References====
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|valign="top"|{{Ref|Ge}}||valign="top"|  S. Gelbart,   "An elementary introduction to the Langlands program"  ''Bull. Amer. Math. Soc.'', '''10'''  (1984)  pp. 177–220  {{MR|0733692}}  {{ZBL|0539.12008}} 
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|valign="top"|{{Ref|RaScSc}}||valign="top"|  M. Rapoport (ed.)  N. Schappacher (ed.)  P. Schneider (ed.), ''Beilinson's conjectures on special values of $L$-functions'', Acad. Press  (1988)  {{MR|0944987}}  {{ZBL|0635.00005}} 
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