  We present a new X-ray luminosity function of flat-spectrum radio quasars
(FSRQs) utilizing the latest {\it Swift}/BAT 105-month X-ray source catalog.
Contrary to previous studies of FSRQs in the X-ray band, using the
luminosity-dependent density evolution model, we find that FSRQs show
evolutionary peaks at $z\sim1-2$ depending on luminosities. Our result is
rather consistent with the evolution of FSRQs seen in the radio and GeV bands,
although the number density is a factor of 5--10 smaller. We further explore
the contribution of FSRQs to the cosmic MeV gamma-ray background radiation. We
find that FSRQs can explain only $\sim3$\% of the observed MeV gamma-ray
background fluxes around 1 MeV, indicating other populations are required.
Future MeV gamma-ray observations will be keys for understanding the origin of
the MeV gamma-ray background radiation.
