Zinc oxide as an optoelectronic device material was studied to utilize its wide band gap of 3.37 eV and high exciton biding energy of 60 meV. Using anti-site nitrogen to generate p-type zinc oxide has shown a deep acceptor level and low solubility. To increase the nitrogen solubility in zinc oxide, group 13 elements (aluminum, gallium, and indium) was co-added to nitrogen. The effect of aluminum on nitrogen solubility in a $3{ imes}3{ imes}2$ zinc oxide super cell containing 72 atoms was investigated using density functional theory with hybrid functionals of Heyd, Scuseria, and Ernzerhof (HSE). Aluminum and nitrogen were substituted for zinc and oxygen sites in the super cell, respectively. The band gap of the undoped super cell was calculated to be 3.36 eV from the density of states, and was in good agreement with the experimentally obtained value. Formation energies of a nitrogen molecule and nitric oxide in the zinc oxide super cell in zinc-rich conditions were lower than those in oxygen-rich conditions. When the number of nitrogen molecules near the aluminum increased from one to four in the super cell, their formation energies decreased to approach the valence band maximum to some degree. However, the acceptor level of nitrogen in zinc oxide with the co-incorporation of aluminum was still deep.