  The polarization of $\tau$ leptons is measured using leptonic and hadronic
$\tau$ lepton decays in Z $\to$ $\tau^+\tau^-$ events in proton-proton
collisions at $\sqrt{s}$ = 13 TeV recorded by CMS at the CERN LHC with an
integrated luminosity of 36.3 fb$^{-1}$. The measured $\tau^-$ polarization at
the Z boson mass pole is $\mathcal{P}_{\tau}$(Z) = $-$0.144 $\pm$ 0.006 (stat)
$\pm$ 0.014 (syst) = $-$0.144 $\pm$ 0.015, in good agreement with the
measurement of the $\tau$ lepton asymmetry parameter of $A_{\tau}$ = 0.1439
$\pm$ 0.0043 = $-\mathcal{P}_{\tau}$(Z) at LEP. The $\tau$ polarization depends
on the ratio of the vector to axial-vector couplings of the $\tau$ leptons in
the neutral current expression, and thus on the effective weak mixing angle
$\sin^{2}\theta_\mathrm{W}^{\text{eff}}$, independently of the Z boson
production mechanism. The obtained value
$\sin^{2}\theta_\mathrm{W}^{\text{eff}}$ = 0.2319 $\pm$ 0.0008 (stat) $\pm$
0.0018 (syst) = 0.2319 $\pm$ 0.0019 is in good agreement with measurements at
e$^+$e$^-$ colliders.
