  We obtain an exact rotating BTZ-like black hole solution by solving the
corresponding gravitational field equations and the bumblebee motion equations
in Einstein-bumblebee gravity theory. Result is presented for the purely radial
Lorentz symmetry violating and can only exist with a linear functional
potential of the bumblebee field. This black hole has two horizons and an
ergosphere which are dependent on the bumblebee coupling constant $\ell$. The
concepts of the area and volume of the horizon should be renewed in this LV
spacetime due to the nontrivial contribution of coupling between the bumblebee
field and the Ricci tensor. Only in this way, the entropy-area relation, first
law of thermodynamics and the Smarr formula can still be constructed. We also
study the AdS/CFT correspondence of this black hole, find that the entropy
product of its inner and outer horizons is universal. So the central charges of
the dual CFT on the boundary can be obtained via the thermodynamic method, and
they can reappear black hole mass and angular momentum in the bulk.
