  We present the phenomenology of antisymmetric rank-2 tensor unparticle
operator ${\cal O}_{\cal U,A}^{\mu\nu}$ with scaling dimension $d_{\cal U}$. We
consider the physical effects of operator $O_{\cal U,A}^{\mu\nu}$ in $Z^0$
boson invisible decays $Z^0 \to {\cal U}$, $Z^0 \to b \bar{b}$ channel, the
electroweak precision observable $S$ parameter, and the muon anomalous magnetic
dipole moment. The $Z^0$ boson invisible decay gives a very stringent
constraint in the $(\Lambda_{\cal U}, M_{\cal U})$ plane, and only small $r
\equiv \Lambda_{\cal U} / M_{\cal U} \lesssim 0.1$ is favored, when
$\Lambda_{\cal U}$ is order of several 100 GeV. When the phenomenological
parameter $\mu$, which parameterizes the scale invariance breaking, goes to 0,
the $S$ parameter and the muon $(g-2)$ diverge for $1 < d_{\cal U} < 2$, while
for non-zero $\mu$, there will be constraints on $(\Lambda_{\cal U}, M_{\cal
U})$ which are more stringent than those obtained from collider experiments.
