  Stationary circularly symmetric solutions of General Relativity with negative
cosmological constant coupled to the Maxwell field are analyzed in three
spacetime dimensions. Taking into account that the fall-off of the fields is
slower than the standard one for a localized distribution of matter, it is
shown that, by virtue of a suitable choice of the electromagnetic Lagrange
multiplier, the action attains a bona fide extremum provided the asymptotic
form of the electromagnetic field fulfills a nontrivial integrability
condition. As a consequence, the mass and the angular momentum become
automatically finite, without the need of any regularization procedure, and
they generically acquire contributions from the electromagnetic field.
Therefore, unlike the higher-dimensional case, it is found that the precise
value of the mass and the angular momentum explicitly depends on the choice of
boundary conditions. It can also be seen that requiring compatibility of the
boundary conditions with the Lorentz and scaling symmetries of the class of
stationary solutions, singles out a very special set of "holographic boundary
conditions" that is described by a single parameter. Remarkably, in stark
contrast with the somewhat pathological behaviour found in the standard case,
for the holographic boundary conditions (i) the energy spectrum of an
electrically charged (rotating) black hole is nonnegative, and (ii) for a fixed
value of the mass, the electric charge is bounded from above.
