what is exactly the difference between the Selberg class and the set of Artin L-functions?

Question

The question is in the title: from what I read in the answer to another question, Artin L-functions are conjecturally cuspidal automorphic L-functions for some algebraic group that can be transfered to $GL_{n}$. On the other hand, elements of the Selberg class are widely believed to be (cuspidal?) automorphic L-functions for $GL_{n}$. So where exactly lies the difference between those two sets of L-functions?
Thanks in advance.

Answer 1

We talk about three rather different but not unrelated conjectures here:

(1) Artin $L$-functions are automorphic $L$-functions;

(2) automorphic $L$-functions belong to the Selberg class;

(3) the Selberg class consists of automorphic $L$-functions.

The three families of $L$-functions occurring here are defined very differently. Artin $L$-functions are defined in terms of Galois representations, automorphic $L$-functions are defined in terms of automorphic representations, while the Selberg class is defined via natural axioms of an analytic nature. Conjectures (1) and (2) are instances of the Langlands conjectures, while (3) strengthens the idea that sufficiently nice analytic properties of a Dirichlet series are always "caused by" an automorphic form (or automorphic representation) behind the Dirichlet series.

Answer 2

The essential answer has already been given but here are a few extra thoughts.

Artin L-functions are defined from a representation of a Galois group on a compex vector space, wheras Hasse--Weil (which are motivic) L-functions are defined by a Galois action on an l-adic one. In the former case, for topological reasons a complex representation of the infinite Galois group Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$ factors through the Galois group of a finite extension $K/\mathbb{Q}$. This is not so for the $l$-adic representations. Hasse--Weil $L$-functions, much like Artin's, are expected to be the $L$-functions of cuspidal automorphic representations.

I notice you query the word "cuspidal" in your question. There are automorphic forms, such as Eisenstein series, which are not cuspidal. Basically this means there is a non-zero constant term in the Fourier expansion. This constant term is closely related to a pole of the $L$-function. To see this, all one needs is Mellin inversion and the residue theorem. Artin $L$-functions can be easily proved meromorphic, as a simple application of the Brauer induction theorem, but they are expected to moreover be holomorphic, and this is not known in generality. This holomorphy is crucial in the application of the so-called converse theorems which would give automorphy of Artin $L$-functions.

Answer 3

As far as I know, the precise conjecture for what you are asking is:

All the elements of the Selberg class that are not Artin L-functions are:

• motivic L-functions of dimension bigger than 0.

• transcendental L-functions

Note that Artin L-functions are the 0-dimensional motivic L-functions.

Also, transcendental L-function is an umbrella term for automorphic L-functions that are not motivic.