  We present a comprehensive experimental and theoretical investigation of the
thermodynamic properties: specific heat, magnetization and thermal expansion in
the vicinity of the field-induced quantum critical point (QCP) around the lower
critical field $H_{c1} \approx 2$\,T in DTN . A $T^{3/2}$ behavior in the
specific heat and magnetization is observed at very low temperatures at
$H=H_{c1}$ that is consistent with the universality class of Bose-Einstein
condensation of magnons. The temperature dependence of the thermal expansion
coefficient at $H_{c1}$ shows minor deviations from the expected $T^{1/2}$
behavior. Our experimental study is complemented by analytical calculations and
Quantum Monte Carlo simulations, which reproduce nicely the measured
quantities. We analyze the thermal and the magnetic Gr\"{u}neisen parameters
that are ideal quantities to identify QCPs. Both parameters diverge at $H_{c1}$
with the expected $T^{-1}$ power law. By using the Ehrenfest relations at the
second order phase transition, we are able to estimate the pressure
dependencies of the characteristic temperature and field scales.
