  INTEGRAL has two sensitive gamma-ray instruments that have detected 46
gamma-ray bursts (GRBs) up to July 2007. We present the spectral, spatial, and
temporal properties of the bursts in the INTEGRAL GRB catalogue using data from
the imager, IBIS, and spectrometer, SPI. Spectral properties of the GRBs are
determined using power-law, Band model and quasithermal model fits to the
prompt emission. Spectral lags, i.e. the time delay in the arrival of
low-energy gamma-rays with respect to high-energy gamma-rays, are measured for
31 of the GRBs. The photon index distribution of power-law fits to the prompt
emission spectra is consistent with that obtained by Swift. The peak flux
distribution shows that INTEGRAL detects proportionally more weak GRBs than
Swift because of its higher sensitivity in a smaller field of view. The all-sky
rate of GRBs above ~0.15 ph cm^-2 s^-1 is ~1400 yr^-1 in the fully coded field
of view of IBIS. Two groups are identified in the spectral lag distribution,
one with short lags <0.75 s (between 25-50 keV and 50-300 keV) and one with
long lags >0.75 s. Most of the long-lag GRBs are inferred to have low redshifts
because of their long spectral lags, their tendency to have low peak energies
and their faint optical and X-ray afterglows. They are mainly observed in the
direction of the supergalactic plane with a quadrupole moment of
Q=-0.225+/-0.090 and hence reflect the local large-scale structure of the
Universe. The rate of long-lag GRBs with inferred low luminosity is ~25% of
Type Ib/c supernovae. Some of these bursts could be produced by the collapse of
a massive star without a supernova or by a different progenitor, such as the
merger of two white dwarfs or a white dwarf with a neutron star or black hole,
possibly in the cluster environment without a host galaxy.
