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The inclusive J/ψ meson production in Pb-Pb collisions at a center-of-mass energy per nucleon-nucleon collision of sNN−−−√ = 5.02 TeV at midrapidity (|y| < 0.9) is reported by the ALICE Collaboration. The measurements are performed in the dielectron decay channel, as a function of event centrality and J/ψ transverse momentum pT, down to pT = 0 GeV/c. The J/ψ mean transverse momentum ⟨pT⟩ and rAA ratio, defined as ⟨p2T⟩PbPb/⟨p2T⟩pp, are evaluated. Both observables show a centrality dependence decreasing towards central (head-on) collisions. The J/ψ nuclear modification factor RAA exhibits a strong pT dependence with a large suppression at high pT and an increase to unity for decreasing pT. When integrating over the measured momentum range pT < 10 GeV/c, the J/ψ RAA shows a weak centrality dependence. Each measurement is compared with results at lower center-of-mass energies and with ALICE measurements at forward rapidity, as well as to theory calculations. All reported features of the J/ψ production at low pT are consistent with a dominant contribution to the J/ψ yield originating from charm quark (re)combination.