Three-dimensional topological Dirac semimetals have recently gained significant attention, since they possess exotic quantum states. When constructing Josephson junctions utilizing these materials as the weak link, the fractional ac Josephson effect emerges in the presence of a topological supercurrent contribution. We investigate the ac Josephson effect in a Dirac semimetal Cd$_3$As$_2$ nanowire using two complementary methods: by probing the radiation spectrum and by measuring Shapiro patterns. With both techniques, we find that conventional supercurrent dominates at all investigated doping levels and that any potentially present topological contribution falls below our detection threshold. The inclusion of thermal noise in a resistively and capacitively shunted junction (RCSJ) model allows us to reproduce the microwave characteristics of the junction. With this refinement, we explain how weak superconducting features can be masked and provide a framework to account for elevated electronic temperatures present in realistic experimental scenarios.
