Inclusive J/$\psi$ production is studied in minimum-bias proton-proton collisions at a centre-of-mass energy of $\sqrt{s}$ = 5.02 TeV by ALICE at the CERN LHC. The measurement is performed at mid-rapidity ($|y| < 0.9$) in the dielectron decay channel down to zero transverse momentum $p_{\rm T}$, using a data sample corresponding to an integrated luminosity of $L_{\rm int} = 19.4 \pm$ 0.4 nb$^{-1}$. The measured $p_{\rm T}$-integrated inclusive J/$\psi$ production cross section is d$\sigma$/d$y$ = 5.64 $\pm$ 0.22 (stat.) $\pm 0.33$ (syst.) $\pm 0.12$ (lumi.) $\mu$b. The $p_{\rm T}$-differential cross section d$^{2} \sigma$/d$p_{\rm T}$d$y$ is measured in the $p_{\rm T}$ range 0$-$10 GeV/$c$ and compared with state-of-the-art perturbative QCD calculations. The J/$\psi$ $\langle p_{\rm T} \rangle$ and $\langle p_{\rm T}^{2} \rangle$ are extracted and compared with results obtained at other collision energies.
$p_{\rm T}$-integrated inclusive J/$\psi$ cross section.
$p_{\rm T}$-differential inclusive J/$\psi$ cross section.
Mean $p_{\rm T}$ square of the inclusive J/$\psi$ spectrum at 5.02 TeV.
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