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Lepton flavour violation in the charged lepton sector is an unambiguous signal of physics beyond the Standard Model. Searches for lepton flavour violation in decays of the $Z$ boson with the ATLAS detector is reported, focusing on decays into an electron or muon and a hadronically decaying $\tau$-lepton, using $pp$ collisions data with a centre-of-mass energy of 13 TeV. Upper limits on the branching ratios of lepton-flavor-violating decays are set at the 95\% confidence level: $\mathcal{B}(Z\to e\tau) < 5.8\times10^{-5}$ and $\mathcal{B}(Z\to \mu\tau) < 2.4\times10^{-5}$. When combined with a previous ATLAS result based on $pp$ collisions data with a centre-of-mass energy of 8 TeV, an upper limit of $\mathcal{B}(Z\to \mu\tau) < 1.3\times10^{-5}$ is obtained.