A search for the $Z\gamma$ decay of the Higgs boson, with $Z$ boson decays into pairs of electrons or muons is presented. The analysis uses proton$-$proton collision data at $\sqrt{s}$ = 13 TeV corresponding to an integrated luminosity of 139 fb$^{-1}$ recorded by the ATLAS detector at the Large Hadron Collider. The observed data are consistent with the expected background with a $p$-value of 1.3%. An upper limit at 95% confidence level on the production cross-section times the branching ratio for $pp\to H\to Z\gamma$ is set at 3.6 times the Standard Model prediction while 2.6 times is expected in the presence of the Standard Model Higgs boson. The best-fit value for the signal yield normalised to the Standard Model prediction is $2.0^{+1.0}_{-0.9}$ where the statistical component of the uncertainty is dominant.
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