  Direct searches for lepton flavor violation in decays of the $Z$ boson with
the ATLAS detector at the LHC are presented. Decays of the $Z$ boson into an
electron or a muon and a hadronically decaying $\tau$ lepton are considered.
The searches are based on a data sample of proton-proton collisions collected
by the ATLAS detector in 2015 and 2016, corresponding to an integrated
luminosity of 36.1 fb$^{-1}$ at a center-of-mass energy of $\sqrt{s} = 13$ TeV.
No significant excess of events above the expected background is observed, and
upper limits on the branching ratios of lepton-flavor-violating decays are set
at the 95% confidence level: $\mathcal{B} (Z\to e\tau) < 5.8 \times 10^{-5}$
and $\mathcal{B} (Z\to \mu\tau) < 2.4 \times 10^{-5}$. This is the first limit
on $\mathcal{B} (Z\to e\tau)$ with ATLAS data. The upper limit on $\mathcal{B}
(Z\to \mu\tau)$ is combined with a previous ATLAS result based on 20.3
fb$^{-1}$ of proton-proton collision data at a center-of-mass energy of
$\sqrt{s} = 8$ TeV and the combined upper limit at 95% confidence level is
$\mathcal{B} (Z\to \mu\tau) < 1.3 \times 10^{-5}$.
