  The angular distributions of lepton pairs in the Drell-Yan process can
provide rich information on the underlying QCD production mechanisms. These
dynamics can be parameterised in terms of a set of frame dependent angular
coefficients, $A_{i=0,\ldots,7}$, which depend on the invariant mass,
transverse momentum, and rapidity of the lepton pair. Motivated by recent
measurements of these coefficients by ATLAS and CMS, and in particular by the
apparent violation of the Lam-Tung relation $A_0-A_2=0$, we perform a precision
study of the angular coefficients at $\mathcal{O}(\alpha_s^3)$ in perturbative
QCD. We make predictions relevant for $pp$ collisions at $\sqrt{s} = 8$ TeV,
and perform comparisons with the available ATLAS and CMS data as well as
providing predictions for a prospective measurement at LHCb. To expose the
violation of the Lam-Tung relationship we propose a new observable
$\Delta^\mathrm{LT} = 1-A_2/A_0$ that is more sensitive to the dynamics in the
region where $A_0$ and $A_2$ are both small. We find that the
$\mathcal{O}(\alpha_s^3)$ corrections have an important impact on the $p_{T,Z}$
distributions for several of the angular coefficients, and are essential to
provide an adequate description of the data. The compatibility of the available
ATLAS and CMS data is reassessed by performing a partial $\chi^2$ test with
respect to the central theoretical prediction which shows that
$\chi^2/N_\mathrm{data}$ is significantly reduced by going from
$\mathcal{O}(\alpha_s^2)$ to $\mathcal{O}(\alpha_s^3)$.
