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This Letter presents the measurement of the fiducial and differentialcross-sections of the electroweak production of a $Z \gamma$ pair inassociation with two jets. The analysis uses 140 fb$^{-1}$ of LHC proton-protoncollision data taken at $\sqrt{s}$=13 TeV recorded by the ATLAS detector duringthe years 2015-2018. Events with a $Z$ boson candidate decaying into either an$e^+e^-$ or $\mu^+ \mu^-$ pair, a photon and two jets are selected. Theelectroweak component is extracted by requiring a large dijet invariant massand a large rapidity gap between the two jets and is measured with an observedand expected significance well above five standard deviations. The fiducial $pp\rightarrow Z \gamma jj$ cross-section for the electroweak production ismeasured to be 3.6 $\pm$ 0.5 fb. The total fiducial cross-section that alsoincludes contributions where the jets arise from strong interactions ismeasured to be $16.8^{+2.0}_{-1.8}$ fb. The results are consistent with theStandard Model predictions. Differential cross-sections are also measured usingthe same events and are compared with parton-shower Monte Carlo simulations.Good agreement is observed between data and predictions.

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Measurement of the cross-sections of the electroweak and total  production of a $Z \gamma$ pair in association with two jets in $pp$  collisions at $\sqrt{s}$ = 13 TeV with the ATLAS detector