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have expanded the Idea-section at L-function in an attempt to transport some actual idea. The main addition are these paragraphs:
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, 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 $L_\sigma$.
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