In this report, we conduct direct numerical simulations (DNS) of weakly spanwise-rotating plane Couette flows at Reynolds number $Re_w = U_wh/\nu= 1300$ (here, $U_w$ is the half the wall velocity difference, and $h$ is half-channel height). A series of simulations with different rotation numbers $Ro = 2\Omega h/U_w$ ($\Omega$ is constant angular velocity component in the spanwise direction) is carried out to investigate the effect of $Ro$ on the flow statistics. Our results show that the flow statistics are affected by the $Ro$, and a "critical" rotation number $Ro^*$ (between $Ro=0.01$ and $Ro=0.05$) is observed, where the kinetic energy of secondary flow contributes about a half of the turbulent kinetic energy, and the mean shear rate at the center line reaches a minimum value. We conjecture that different mechanisms should exist around $Ro^*$, and will be investigated further.