  Recent lattice studies exhibit infrared finite effective QCD charges.
Corresponding gluon propagator in Landau gauge is finite and nonzero,
suggesting a mechanism of dynamical gluon mass generation is in the operation.
In this paper, the analytical continuation of the Euclidean (spacelike) Pinch
Technique-Background Field Method (PT-BFM) solution of Schwinger-Dyson equation
for gluon propagator to the timelike region of $q^2$ is found. We found the
continuation numerically showing good agreement with a generalized Lehman
representation for small Schwinger coupling. The associate non-positive
spectral function has an unexpected behavior. Albeit infrared Euclidean space
solution naively suggests like single scale "massive" propagator, the obtained
spectrum of gluon propagator does not correspond to the delta function at
single scale $q=m$, instead more possible singularities are generated. The
pattern depends on the details of assumed Schwinger mechanism: for stronger
coupling there are few maxima and minima which appear at the scale $\Lambda$,
while for perturbatively small Schwinger coupling the spectral function shows
up two narrow peaks: particle and ghost excitation, which have mutually
opposite signs.
