  We present results from full general relativistic three-dimensional
hydrodynamics simulations of stellar core collapse of a 70 M$_\odot$ star with
spectral neutrino transport. To investigate the impact of rotation on
non-axisymmetric instabilities, we compute three models by parametrically
changing the initial strength of rotation. The most rapidly rotating model
exhibits a transient development of the low-$T/|W|$ instability with one-armed
spiral flow at the early postbounce phase. Subsequently, the two-armed spiral
flow appears, which persists during the simulation time. The moderately
rotating model also shows the growth of the low-$T/|W|$ instability, but only
with the two-armed spiral flow. In the nonrotating model, a vigorous activity
of the standing accretion-shock instability (SASI) is only observed. The SASI
is first dominated by the sloshing mode, which is followed by the spiral SASI
until the black hole formation. We present a spectrogram analysis of the
gravitational waves (GWs) and neutrinos, focusing on the time correlation. Our
results show that characteristic time modulations in the GW and neutrino
signals can be linked to the growth of the non-axisymmetric instabilities. We
find that the degree of the protoneutron star (PNS) deformation, depending upon
which modes of the non-axisymmetric instabilities develop, predominantly
affects the characteristic frequencies of the correlated GW and neutrino
signals. We point out that these signals would be simultaneously detectable by
the current-generation detectors up to $\sim10$ kpc. Our findings suggest that
the joint observation of GWs and neutrinos is indispensable for extracting
information on the PNS evolution preceding the black hole formation.
