  We present a lattice calculation of the masses and decay constants of
$D_{(s)}^*$ and $B_{(s)}^*$ mesons using the gauge configurations produced by
the European Twisted Mass Collaboration (ETMC) with $N_f = 2 + 1 + 1$ dynamical
quarks at three values of the lattice spacing $a \sim (0.06 - 0.09)$ fm. Pion
masses are simulated in the range $M_\pi \simeq (210 - 450)$ MeV, while the
strange and charm sea-quark masses are close to their physical values. We
compute the ratios of vector to pseudoscalar masses and decay constants for
various values of the heavy-quark mass $m_h$ in the range $0.7
m_c^{\mathrm{phys}} \lesssim m_h \lesssim 3 m_c^{\mathrm{phys}}$. In order to
reach the physical b-quark mass, we exploit the Heavy Quark Effective Theory
prediction that, in the static limit of infinite heavy-quark mass, the
considered ratios are equal to one. At the physical point our results are:
$M_{D^*} / M_{D} = 1.0769(79)$, $M_{D^*_{s}} / M_{D_{s}} = 1.0751(56)$, $
f_{D^*} / f_{D} = 1.078(36),$ $f_{D^*_{s}} / f_{D_{s}} = 1.087(20)$, $M_{B^*} /
M_{B} = 1.0078(15)$, $M_{B^*_{s}} / M_{B_{s}} = 1.0083(10)$, $f_{B^*} / f_{B} =
0.958(22)$ and $f_{B^*_{s}} / f_{B_{s}} = 0.974(10)$. Combining them with the
experimental values of the pseudoscalar meson masses (used as input to fix the
quark masses) and the values of the pseudoscalar decay constants calculated by
ETMC, we get: $M_{D^*} = 2013(14)~\mathrm{MeV}$, $ M_{D_s^*} =
2116(11)~\mathrm{MeV}$, $f_{D^*} = 223.5(8.4)~\mathrm{MeV}$, $f_{D_s^*} =
268.8(6.6)~\mathrm{MeV}$, $M_{B^*} = 5320.5(7.6)~\mathrm{MeV}$, $M_{B_s^*} =
5411.36(5.3)~\mathrm{MeV}$, $f_{B^*} = 185.9(7.2)~\mathrm{MeV}$ and $f_{B_s^*}
= 223.1(5.4)~\mathrm{MeV}$.
