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Up to the third post-Newtonian (3PN) order, we compute (i) the current-typequadrupole moment of (non-spinning) compact binary systems, as well as (ii) thecorresponding gravitational-wave mode $(2,1)$ (constituting a 3.5PN correctionin the waveform). Moreover, at this occasion, (iii) we recompute and confirmthe previous calculation of the mass-type octupole moment to 3PN order. Theultra-violet (UV) divergences due to the point-like nature of the source aretreated by means of dimensional regularization. This entails generalizing thedefinition of the irreducible mass and current multipole moments of an isolatedPN source in $d$ spatial dimensions. In particular, we find that the currenttype moment has the symmetry of a particular mixed Young tableau and that, inaddition, there appears a third type of moment which is however inexistent in 3spatial dimensions.

The current-type quadrupole moment and gravitational-wave mode $(\ell,m)=(2,1)$ of compact binary systems at the third post-Newtonian order