Automorphic L-function

Automorphic ${\displaystyle L}$-functions should have the following properties (which have been proved in some cases but are still conjectural in other cases).

The L-function ${\displaystyle L(s,\pi ,r)}$ should be a product over the places ${\displaystyle v}$ of ${\displaystyle F}$ of local ${\displaystyle L}$ functions.

${\displaystyle L(s,\pi ,r)=\prod _{v}L(s,\pi _{v},r_{v})}$

Here the automorphic representation ${\displaystyle \pi =\otimes \pi _{v}}$ is a tensor product of the representations ${\displaystyle \pi _{v}}$ of local groups.

The L-function is expected to have an analytic continuation as a meromorphic function of all complex ${\displaystyle s}$, and satisfy a functional equation

${\displaystyle L(s,\pi ,r)=\epsilon (s,\pi ,r)L(1-s,\pi ,r^{\lor })}$

where the factor ${\displaystyle \epsilon (s,\pi ,r)}$ is a product of "local constants"

${\displaystyle \epsilon (s,\pi ,r)=\prod _{v}\epsilon (s,\pi _{v},r_{v},\psi _{v})}$

almost all of which are 1.

Godement & Jacquet (1972) constructed the automorphic L-functions for general linear groups with r the standard representation (so-called standard L-functions) and verified analytic continuation and the functional equation, by using a generalization of the method in Tate's thesis. Ubiquitous in the Langlands Program are Rankin-Selberg products of representations of GL(m) and GL(n). The resulting Rankin-Selberg L-functions satisfy a number of analytic properties, their functional equation being first proved via the Langlands–Shahidi method.

In general, the Langlands functoriality conjectures imply that automorphic L-functions of a connected reductive group are equal to products of automorphic L-functions of general linear groups. A proof of Langlands functoriality would also lead towards a thorough understanding of the analytic properties of automorphic L-functions.

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