Lightning is an example of an electric discharge in air. These electric discharges can heat the air to very high temperatures and become an intense source of ultraviolet light that can photoionize the ambient air. This produces energetic free electrons in the air, making it conductive to electric current and thus may change the dynamics of the discharge. This has not been included in previous theoretical studies and numerical simulations. A detailed development of a numerical simulation is first presented without photoionization, reproducing previous works. When photoionization was included, there was only a small difference in the radius of the discharge over time. The position of the shock wave and its peak pressure over time were also nearly the same in both cases. Photoionization, however, did result in a decrease of the arcs core temperature and an increase in its electrical conductivity. Since the measurement of the core temperature is difficult, the only accessible measurement is electrical conductivity, which can evaluate the importance of photoionization.