  We consider the gluon propagator in the Landau gauge at low spacelike momenta
and with the dressing function $Z(Q^2)$ at the two-loop order. We incorporate
the nonperturbative effects by making the (noninteger) powers of the QCD
coupling in the dressing function $Z(Q^2)$ analytic (holomorphic) via the
Fractional Analytic Perturbation Theory (FAPT) model, and simultaneously
introducing the gluon dynamical mass in the propagator as motivated by the
previous analyses of the Dyson-Schwinger equations. The obtained propagator has
behavior compatible with the unquenched lattice data ($N_f=2+1$) at low
spacelike momenta $0.4 \ {\rm GeV} < Q \lesssim 10$ GeV. We conclude that the
removal of the unphysical Landau singularities of the powers of the coupling
via the (F)APT prescription, in conjunction with the introduction of the
dynamical mass $M \approx 0.62$ GeV of the gluon, leads to an acceptable
behavior of the propagator in the infrared regime.
