  We study properties of 2+1-flavor QCD in the imaginary chemical potential
region by using two approaches. One is a theoretical approach based on QCD
partition function, and the other is a qualitative one based on the
Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model. In the theoretical
approach, we clarify conditions imposed on the imaginary chemical potentials
$\mu_{f}=i\theta_{f}T$ to realize the Roberge-Weiss (RW) periodicity. We also
show that the RW periodicity is broken if anyone of $\theta_{f}$ is fixed to a
constant value. In order to visualize the condition, we use the PNJL model as a
model possessing the RW periodicity, and draw the phase diagram as a function
of $\theta_{u}=\theta_{d}\equiv \theta_{l}$ for two conditions of
$\theta_{s}=\theta_{l}$ and $\theta_{s}=0$. We also consider two cases,
$(\mu_{u},\mu_{d},\mu_{s}) =(i\theta_{u}T,iC_{1}T,0)$ and
$(\mu_{u},\mu_{d},\mu_{s})=(iC_{2}T,iC_{2}T,i\theta_{s}T)$; here $C_{1}$ and
$C_{2}$ are dimensionless constants, whereas $\theta_{u}$ and $\theta_{s}$ are
treated as variables. For some choice of $C_{1}$ ($C_{2}$), the number density
of up (strange) quark becomes smooth in the entire region of $\theta_{u}$
($\theta_{s}$) even in high $T$ region.
