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.



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


