  The ratio of branching fractions ${\cal{R}}(D^{*-})\equiv {\cal{B}}(B^0 \to
D^{*-} \tau^+ \nu_{\tau})/{\cal{B}}(B^0 \to D^{*-} \mu^+\nu_{\mu})$ is measured
using a data sample of proton-proton collisions collected with the LHCb
detector at center-of-mass energies of 7 and 8 Tev, corresponding to an
integrated luminosity of 3$~$fb$^{-1}$. For the first time ${\cal{R}}(D^{*-})$
is determined using the $\tau$ lepton decays with three charged pions in the
final state. The $B^0 \to D^{*-} \tau^+\nu_{\tau}$ yield is normalized to that
of the $B^0\to D^{*-} \pi^+\pi^-\pi^+$ mode, providing a measurement of
${\cal{B}}(B^0\to D^{*-}\tau^+\nu_{\tau})/{\cal{B}}(B^0\to
D^{*-}\pi^+\pi^-\pi^+) = 1.97 \pm 0.13 \pm 0.18$, where the first uncertainty
is statistical and the second systematic. The value of ${\cal{B}}(B^0 \to
D^{*-} \tau^+ \nu_{\tau}) = (1.42 \pm 0.094 \pm 0.129 \pm 0.054)\% $ is
obtained, where the third uncertainty is due to the limited knowledge of the
branching fraction of the normalization mode. Using the well-measured branching
fraction of the $B^0 \to D^{*-} \mu^+\nu_{\mu}$ decay, a value of
${\cal{R}}(D^{*-}) = 0.291 \pm 0.019 \pm 0.026 \pm 0.013$ is established, where
the third uncertainty is due to the limited knowledge of the branching
fractions of the normalization and $B^0\to D^{*-}\mu^+\nu_{\mu}$ modes. This
measurement is in agreement with the Standard Model prediction and with
previous results.
