  In this paper we investigate the phenomenology of the U(1) gauge symmetry for
right-handed fermions, where three right-handed neutrinos are introduced for
anomalies cancellation. Constraints on the new gauge boson $Z_{\mathbf{R}}$
arising from $Z-Z^\prime$ mixing as well as the upper bound of $Z^\prime$
production cross section in di-lepton channel at the LHC are presented. We
further study the neutrino mass generation and the phenomenology of
$Z_{\mathbf{R}}$-portal dark matter in this model. The lightest right-handed
neutrino can be the cold dark matter candidate stabilized by a $Z_2$ flavor
symmetry. Our results show that active neutrino masses can be generated via the
modified type-II seesaw mechanism; right-handed neutrino dark matter is
available only for its mass at near the resonant regime of the SM Higgs and/or
the new bosons; constraint from the dilepton search at the LHC is stronger than
that from the $Z-Z^\prime$ mixing only for $g_\mathbf{R}<0.121$, where
$g_\mathbf{R}$ is the new gauge coupling.
