From Book 1 of the Iliad (Buckley translation):
But the Grecian youths throughout the day were appeasing the god by song, chanting the joyous paean, hymning the Far-Darter, and he was delighted in his mind as he listened.
According to some, singing a paean was common practice for the Greeks before entering battle and after victory. It appears that talking about the paean as if it were a specific song is a mistake, for the text certainly varied over time (at least 8 centuries) and probably depending on local religious practice. In the 4th century B.C., the song is probably commonly addressed to Apollo as the giver of victory. Certainly sung by both sides on many battlefields (each their own version?) during the many wars between Greek cities and federations of the period.