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 $GL_2( \mathbb{Q} )$.

$\mathcal S_0=\{1\}$

$\mathcal S_1 =\{\zeta(s),L(s,\chi)\}$ $\chi$ 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?