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We give a full description of the functions $F$ of degree 2 and conductor 1in the general framework of the extended Selberg class. This is performed bymeans of a new numerical invariant $\chi_F$, which is easily computed from thedata of the functional equation. We show that the value of $\chi_F$ gives aprecise description of the nature of $F$, thus providing a sharp form of theclassical converse theorems of Hecke and Maass. In particular, our resultconfirms, in the special case under consideration, the conjecture that thefunctions in the Selberg class are automorphic $L$-functions.

Classification of L-functions of degree 2 and conductor 1