Given a group G, the conjugator length function measures, for two conjugate elements g, g' of length at most n, the minimal length of a conjugator, i.e. of some h such that hg=g'h. The conjugator length has been determined for numerous classes of groups. In this talk, we discuss which functions appear as conjugator length functions. Our construction factors through a class of groups by Hall, which has recently proven to be very expressive in all manners of conjugacy and residual finiteness.