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