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gave automorphic L-function a minimum of an Idea-section, presently it reads as follows:
An automorphic L-function $L_\pi$ is an L-function built from an automorphic representation $\pi$, in nonabelian generalization of how a Dirichlet L-function $L_\chi$ is associated to a Dirichlet character $\chi$ (which is an automorphic form on the (abelian) idele group).
In analogy to how Artin reciprocity implies that to every 1-dimensional Galois representation $\sigma$ there is a Dirichlet character $\chi$ such that the Artin L-function $L_\sigma$ equals the Dirichlet L-function $L_\chi$, so the conjectured Langlands correspondence says that to every $n$-dimensional Galois representation $\sigma$ there is an automorphic representation $\pi$ such that the automorphic L-function $L_\pi$ equals the Artin L-function $L_\sigma$.
added at the beginning of the Idea section briefly the conceptual statement that passing from an automorphic form to its automorphic L-function is the generalization of the Mellin transform taking an automorphic form to its zeta function.
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