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.
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