Recently, we reported the discovery of enormous negative viscosity of a nematic liquid crystal in the presence of turbulence induced by ac electric fields, which enabled us to observe unique phenomena related to the negative viscosity, such as spontaneous shear flow, hysteresis in flow curves, and self-oscillation [Orihara et al., Phys. Rev. E 99, 012701 (2019)10.1103/PhysRevE.99.012701]. In the present paper, we report the rheological properties of another nematic liquid crystal, which is a homologue of the previous one. The properties of the present liquid crystal are strongly dependent on electrical conductivity. Three samples with different conductivities were prepared by changing the amount of an ionic dopant. It was found that the lowest-conductivity sample without dopant shows no negative viscosity whereas the other ion-doped samples exhibit negative viscosity with strong dependence on the frequency of the ac electric field, consistent with microscopic observations. Phase diagrams of the negative- and positive-viscosity states in the amplitude and frequency plane are constructed to show the conductivity effect. Furthermore, we propose a model to reproduce another type of self-oscillation found in the present study.