  We show numerically that the Abelian Higgs field equations in the background
of a four-dimensional rotating charged black string have vortex solutions.
These solutions which have axial symmetry show that the rotating black string
can support the Abelian Higgs field as hair. We find that one encounters with
an electric field coupled to the Higgs scalar field for the case of rotating
black string. This electric field is due to an electric charge per unit length,
which increases as the rotation parameter becomes larger. We also find that the
vortex thickness decreases as the rotation parameter grows up. Finally we
consider the self-gravity of the Abelian Higgs field and show that the effect
of the vortex is to induce a deficit angle in the metric under consideration
which decreases as the rotation parameter increases.
