  The Lam-Tung relation breaking coefficient $A_{\mathrm{LT}} = A_0 - A_2$ in
the Drell-Yan dilepton angular distributions in the $Z^0$ boson mass region at
the LHC is analyzed in the $k_T$-factorization approach. This observable was
recently measured with high precision by ATLAS collaboration. Within the
$k_T$-factorization approach we perform an approximate ${\cal
O}(\alpha_{\mathrm{em}}\alpha_s^2)$ calculation of the off-shell parton hard
matrix elements in which we include the leading tree level contributions of
valence quarks and off-shell gluons: the $q_{\mathrm{val}} g^* \to qZ^0$
channel and the $g^*g^* \to q\bar q Z^0$ channel. The resulting
$A_{\mathrm{LT}}$ exhibits high sensitivity to the gluon transverse momentum
distribution (TMD). Several gluon TMDs are probed derived from the CCFM and
BFKL evolution equations, and given by QCD-inspired phenomenological
parameterizations. The ATLAS data favor a simple "Weizs\"acker-Williams" (WW)
hard gluon TMD with the asymptotic behavior of one-gluon exchange at large
gluon transverse momenta and moderate $x$. It is verified that the proposed
approach with the WW gluon TMD describes well also the $A_0$ and $A_2$ angular
coefficients at the $Z^0$ peak, as well as the Drell-Yan dilepton mass
distribution at lower masses. We conclude that inclusion of gluon transverse
momentum effects improves description of the angular distributions of Drell-Yan
dileptons and that the Drell-Yan scattering provides an excellent probe of the
parton TMDs.
