Abstract: A purely analytical approach to non-perturbative quantum chromodynamics is discussed. The exact, gauge-fixed, Faddeev-Popov Lagrangian of Yang-Mills theory is studied by the screened massive expansion that emerges from a mere change of the expansion point of ordinary perturbation theory. The gluon propagator has gauge-invariant complex conjugated poles that might give a direct dynamical proof of gluon confinement. Their genuine nature is discussed. Because of BRST symmetry, the analytic properties and the poles are shown to play a central role in the optimization of the expansion, which becomes a very predictive and ab initio tool. While in excellent agreement with the lattice data in the Euclidean space, the expansion provides valuable information in sectors that are not easily explored on the lattice, such as Minkowski space and a generic covariant gauge. Moreover, even in the Euclidean space, the method gives a lattice-independent estimate of the running coupling in the continuum limit.