  We study the superconducting gap function of Sr$_2$RuO$_4$. By solving the
linearized Eliashberg equation with a correlated pairing vertex extracted from
a dynamical mean-field calculation we identify the dominant pairing channels.
An analysis of the candidate gap functions in orbital and quasiparticle band
basis reveals that an inter-orbital singlet pairing of even parity is in
agreement with experimental observations. It reconciles in particular the
occurrence of a two-component order parameter with the presence of line-nodes
of quasiparticles along the c-axis in the superconducting phase. The strong
angular dependence of the gap along the Fermi surface is in stark contrast to
its quasi-locality when expressed in the orbital basis. We identify local
inter-orbital spin correlations as the driving force for the pairing and thus
reveal the continuation of Hund's physics into the superconducting phase.
