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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. This situation holds also in the case of the extremal black string. The main point which we find is the existence of an electric field associated to the Higgs field for the case of rotating black string. This Higgs electric field is due to a Higgs electric charge per unit length, which increases as the rotation parameter becomes larger. We also 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.
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