The first measurement of the $Z$ boson production cross-section at centre-of-mass energy $\sqrt{s} = 5.02\,$TeV in the forward region is reported, using $pp$ collision data collected by the LHCb experiment in year 2017, corresponding to an integrated luminosity of $100 \pm 2\,\rm{pb^{-1}}$. The production cross-section is measured for final-state muons in the pseudorapidity range $2.0<\eta<4.5$ with transverse momentum $p_{\rm{T}}> 20\,\rm{GeV/}\it{c}$. The integrated cross-section is determined to be
\[ \sigma_{Z \rightarrow \mu^{+}\mu^{-}} = 39.6 \pm 0.7\,(\rm{stat}) \pm 0.6\,(\rm{syst}) \pm 0.8\,(\rm{lumi}) \ \rm{pb} \]
for the di-muon invariant mass in the range $60
Systematic uncertainties on the single differential cross-sections in bins of $y^{Z}$, presented in percentage.
Systematic uncertainties on the single differential cross-sections in bins of $p_{T}^{Z}$, presented in percentage.
Systematic uncertainties on the single differential cross-sections in bins of $\phi_{\eta}^{*}$, presented in percentage.
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