This study shows that for weakly unstable Rossby wave regimes, the classical dispersion relation, which relates complex corrections to frequency and wavenumber, has an exception. In the space of wave vectors, there is a point at which the zonal component of the group velocity of Rossby waves becomes zero. This point forms the second branch of the Rossby neutral wave frequency curve, characterized by complex wavenumbers. Thus, the paradigm according to which a complex frequency corresponds to a complex wavenumber has exceptions in the case of Rossby waves.