  In a rigidly-rotating magnetohydrodynamic (MHD) system with convective
turbulence, a large-scale dynamo, categorized as the $\alpha^2$-type, can be
excited when the spin rate is large enough. In this paper, the rotational
dependence of the $\alpha^2$-type dynamo and the cause of it are explored by
mean-field (MF) dynamo models coupled with direct numerical simulations (DNSs)
of MHD convections. Bearing the application to the solar/stellar dynamo in
mind, we adopt a strongly-stratified polytrope as a model of the convective
atmosphere. Our DNS models show that the $\alpha^2$-type dynamo is excited when
${\rm Ro} \lesssim 0.1$ where ${\rm Ro}$ is the Rossby number defined with the
volume-averaged mean convective velocity. From the corresponding MF models, we
demonstrate that the rotational dependence of the $\alpha^2$-type dynamo is
mainly due to the change in the magnitude of the turbulent magnetic diffusion.
With increasing the spin rate, the turbulent magnetic diffusion weakens while
the $\alpha$-effect remains essentially unchanged over the convection zone,
providing the critical point for the excitation of the large-scale dynamo. The
${\rm Ro}$-dependence of the stellar magnetic activity observable in the cool
star is also discussed from the viewpoint of the rotational dependence of the
turbulent electro-motive force. Overall our results suggest that, to get a
better grasp of the stellar dynamo activity and its ${\rm Ro}$-dependence, it
should be quantified how the convection velocity changes with the stellar spin
rate with taking account of the rotational quenching and the Lorentz force
feedback from the magnetic field on the convective turbulence.
