  Fermi has provided the largest sample of {\gamma}-ray selected blazars to
date. In this work we use a complete sample of FSRQs detected during the first
year of operation to determine the luminosity function (LF) and its evolution
with cosmic time. The number density of FSRQs grows dramatically up to redshift
\sim0.5-2.0 and declines thereafter. The redshift of the peak in the density is
luminosity dependent, with more luminous sources peaking at earlier times; thus
the LF of {\gamma}-ray FSRQs follows a luminosity-dependent density evolution
similarly to that of radio-quiet AGN. Also using data from the Swift Burst
Alert Telescope we derive the average spectral energy distribution of FSRQs in
the 10 keV-100 GeV band and show that there is no correlation of the peak
{\gamma}-ray luminosity with {\gamma}-ray peak frequency. The coupling of the
SED and LF allows us to predict that the contribution of FSRQs to the Fermi
isotropic {\gamma}-ray background is 9.3(+1.6/-1.0) (\pm3% systematic
uncertainty) in the 0.1-100GeV band. Finally we determine the LF of unbeamed
FSRQs, finding that FSRQs have an average Lorentz factor of {\gamma} =
11.7(+3.3/-2.2), that most are seen within 5\circ of the jet axis, and that
they represent only ~0.1 % of the parent population.
