  The spin-singlet state $\eta_{_{c2}}(^1D_{_2})$ has not been discovered in
experiment and it is the only missing low-excited $D$-wave charmonium, so in
this paper, we like to study its properties. Using the Bethe-Salpeter equation
method, we obtain its mass as $3828.2$ MeV and its electromagnetic decay widths
as $\Gamma[\eta_{_{c2}}(1D)\rightarrow h_{_{c}}(1P)\gamma]=284$ keV,
$\Gamma[\eta_{_{c2}}(1D)\rightarrow J/\psi\gamma]=1.04$ keV,
$\Gamma[\eta_{_{c2}}(1D)\rightarrow\psi(2S)\gamma]=3.08$ eV, and
$\Gamma[\eta_{_{c2}}(1D)\rightarrow\psi(3770)\gamma]=0.143$ keV. We estimate
its full width to be about $366$ keV, and point out that the electromagnetic
decay partial width is very sensitive to its mass and show the variation of the
width along with the mass in the range of $3800\sim3872$ MeV. In our
calculation, the emphasis is put on the relativistic corrections. Our results
show that $\eta_{_{c2}}\rightarrow h_{_{c}}\gamma$ is the non-relativistic $E1$
transition dominated $E1+M2+E3$ decay, and $\eta_{_{c2}}\rightarrow \psi\gamma$
is the $M1+E2+M3+E4$ decay but the relativistic $E2$ transition contributes the
most.
