  In this work we study the r-mode instability windows and the gravitational
wave signatures of neutron stars in the slow rotation approximation using the
equation of state obtained from the density dependent M3Y effective
interaction. We consider the neutron star matter to be $\beta$-equilibrated
neutron-proton-electron matter at the core with a rigid crust. The fiducial
gravitational and viscous timescales, the critical frequencies and the time
evolutions of the frequencies and the rates of frequency change are calculated
for a range of neutron star masses. We show that the young and hot rotating
neutron stars lie in the r-mode instability region. We also emphasize that if
the dominant dissipative mechanism of the r-mode is the shear viscosity along
the boundary layer of the crust-core interface, then the neutron stars with low
$L$ value lie in the r-mode instability region and hence emit gravitational
radiation.
