  We theoretically investigate superfluid properties of a two-band gas of
$^{173}$Yb Fermi atoms with an orbital Feshbach resonance (OFR). To describe
the BCS-BEC crossover region, we include superfluid fluctuations caused by
inter-band and intra-band pairing interactions associated with OFR, by
extending the strong-coupling theory developed by Nozi\`eres and Schmitt-Rink
to the two-band case below the superfluid phase transition temperature;
however, effects of an experimentally inaccessible deep bound state are
removed, to model a real $^{173}$Yb Fermi gas near OFR. We show that the
condensate fraction in the upper closed channel gradually becomes smaller than
that in the lower open channel, as one moves from the strong- to the
weak-coupling regime, because the OFR-pairing mechanism tunes the interaction
strengths by adjusting the energy difference between the two bands. However,
even when the closed-channel band is much higher in energy than the
open-channel band in the weak-coupling regime, the magnitude of the superfluid
order parameter in the closed channel is found to be still comparable to that
in the open channel. As the reason for this, we point out a pair-tunneling
effect by the OFR-induced inter-band interaction. Besides these superfluid
quantities, we also examine collective modes, such as the Goldstone mode,
Schmid (Higgs) mode, as well as Leggett mode, to clarify how they appear in the
spectral weights of pair-correlation functions in each band. Since the
realization of a multi-band superfluid Fermi gas is a crucial issue in cold
Fermi gas physics, our results would contribute to the basic understanding of
this type of Fermi superfluid in the BCS-BEC crossover region.
