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A paper by Kaczorowski & Perelli arXiv:1207.2312 dealing with the elements of the Selber class with degree two suggests that $S_2$ coincides with the automorphic l-functions over $GL_2( \mathbb{Q} )$.

EDIT. We know from [1][2][3]:

$\mathcal S_0=\{1\}$

$\mathcal S_1 =\{\zeta(s),L(s,\chi)\}$ $\chi$ primitive plus shifts.

$\mathcal S_2 =\{\zeta^2(s),L(s,\chi_1)L(s,\chi_2),L_f(s),\zeta_K(s)\}$ primitives, shifts (list only conjecturally complete)

Can be those classification results proved for automorphic representation on $GL_n( \mathbb{Q} )$? Improved perhaps for $\mathcal S_2$? References? Conjectures for arbitrary n?

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I think your question is too broad, at least it is not clear to me what you are really asking. At any rate, if the Ramanujan-Selberg conjecture is true, then the $L$-function of an automorphic representation of $GL_n$ over $\mathbb{Q}$ (with unitary central character) belongs to the Selberg class. Moreover, it is believed that these are all the elements of the Selberg class, hence in particular every element of the Selberg class should satisy the Generalized Riemann Hypothesis (I think this was Selberg's original motivation). The $L$-functions coming from cuspidal representations should agree with the set of primitive elements in the Selberg class. The extended Selberg class is definitely larger, because it is closed under complex linear combinations. You can find further information in the surveys of Perrelli, e.g. in this one.

Note that the Ramanujan-Selberg conjecture is only known in very special cases (e.g. for certain automorphic representations of cohomological type). On the other hand, the relevant axiom of the Selberg class can definitely be weakened without really enlarging the class (as far as we know), and then the principal automorphic $L$-functions mentioned above provably belong to the modified class.

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  • $\begingroup$ A concrete example: the next step in the Kaczorowski/Perelli approach to the degree conjecture would be to classify the Selberg class of degree 2. If the list here (arXiv:1207.2312, fourth paragraph) includes all known automorphic forms of degree 2, you can try to prove the manageably hard result that the list is complete. If that is not the case, you need a Ramanujan-Selberg-type result for the additional forms, which falls outside the scope of Selberg's class theory. $\endgroup$
    – Myshkin
    Commented May 16, 2014 at 12:54
  • $\begingroup$ Another example: why do automorphic forms with (conjectural or not) functional equation of noninteger degree fail to exist? Ideas in this direction could be translated in arguments you might hope to prove once you assume all (or some) of Selberg's axioms. $\endgroup$
    – Myshkin
    Commented May 16, 2014 at 12:56
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    $\begingroup$ @Myshkin: For your second question, it is known that the $L$-function of an automorphic representation on $GL_n$ over $\mathbb{Q}$ has degree $n$. For automorphic representations on other groups, the situation is not so clear, but according to the Langlands conjectures these do not lead to a wider class of $L$-functions. $\endgroup$
    – GH from MO
    Commented May 16, 2014 at 13:01

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