  Using the operator product expansion we show that the $O(1/p^2)$ correction
to the perturbative expressions for the gluon propagator and the strong
coupling constant resulting from lattice simulations in the Landau gauge are
due to a non-vanishing vacuum expectation value of the operator $A^\mu A_\mu$.
This is done using the recently published Wilson coefficients of the identity
operator computed to third order, and the subdominant Wilson coefficient
computed in this paper to the leading logarithm. As a test of the applicability
of OPE we compare the $<A^\mu A_\mu>$ estimated from the gluon propagator and
the one from the coupling constant in the flavourless case. Both agree within
the statistical uncertainty: $\sqrt{< A^\mu A_\mu >} \simeq 1.64(15)$ GeV.
Simultaneously we fit $\Lams = $ 233(28) MeV in perfect agreement with previous
lattice estimates. When the leading coefficients are only expanded to two
loops, the two estimates of the condensate differ drastically. As a consequence
we insist that OPE can be applied in predicting physical quantities only if the
Wilson coefficients are computed to a high enough perturbative order.
