A bstract The polarization of τ leptons is measured using leptonic and hadronic τ lepton decays in Z → τ + τ − events in proton-proton collisions at $$ \sqrt{s} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mi>s</mml:mi> </mml:msqrt> </mml:math> = 13 TeV recorded by CMS at the CERN LHC with an integrated luminosity of 36.3 fb − 1 . The measured τ − lepton polarization at the Z boson mass pole is $$ {\mathcal{P}}_{\tau}\left(\textrm{Z}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>τ</mml:mi> </mml:msub> <mml:mfenced> <mml:mi>Z</mml:mi> </mml:mfenced> </mml:math> = −0.144 ± 0.006 (stat) ± 0.014 (syst) = −0.144 ± 0.015, in good agreement with the measurement of the τ lepton asymmetry parameter of A τ = 0.1439 ± 0.0043 = $$ -{\mathcal{P}}_{\tau}\left(\textrm{Z}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>τ</mml:mi> </mml:msub> <mml:mfenced> <mml:mi>Z</mml:mi> </mml:mfenced> </mml:math> at LEP. The τ lepton polarization depends on the ratio of the vector to axial-vector couplings of the τ leptons in the neutral current expression, and thus on the effective weak mixing angle sin 2 $$ {\theta}_{\textrm{W}}^{\textrm{eff}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>θ</mml:mi> <mml:mi>W</mml:mi> <mml:mi>eff</mml:mi> </mml:msubsup> </mml:math> , independently of the Z boson production mechanism. The obtained value sin 2 $$ {\theta}_{\textrm{W}}^{\textrm{eff}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>θ</mml:mi> <mml:mi>W</mml:mi> <mml:mi>eff</mml:mi> </mml:msubsup> </mml:math> = 0.2319 ± 0 . 0008(stat) ± 0 . 0018(syst) = 0 . 2319 ± 0 . 0019 is in good agreement with measurements at e + e − colliders.