  In this paper, we make a comprehensive study of the properties of a gapped
Dirac semimetal model, which was originally proposed in the magnetoinfrared
spectroscopy measurement of ZeTe$_5$, and includes both the linear and
parabolic dispersions in all three directions. We find that, depending on the
band inversion parameters, $\zeta'$ and $\zeta_z'$, the model can support three
different phases: the single Dirac point (DP) phase, the double DPs phase and
the Dirac ring phase. The three different phases can be distinguished by their
low-energy features in the density of states (DOS) and optical conductivity. At
high energy, both the DOS and optical conductivity exhibit power-law like
behaviors, with the asymptotic exponents depending heavily on the signs of
$\zeta'$ and $\zeta_z'$. Moreover, the thumb-of-rule formula between the DOS
and optical conductivity is satisfied only when $(\zeta',\zeta_z')>0$. The
implications of our results for experiments are discussed.
