Nitrogen-rich compounds containing polynitrogen are attractive candidates for high-energy-density materials. In this work, using first-principles calculations and a particle swarm optimization structural search method, four novel nitrogen-rich structures are predicted at high pressures, i.e., two ZnN3 phases with the same space group P-1 (low-pressure phase LP-ZnN3 and high-pressure phase HP-ZnN3), Cmm2-ZnN5 and Pcc2-ZnN6¬, the energy density are estimated to be 1.41 kJ/g, 1.88 kJ/g, 4.07 kJ/g, and 2.60 kJ/g, respectively. LP-ZnN3 (54~72 GPa) and HP-ZnN3 (above 72 GPa) have the lowest enthalpies in all known ZnN3 phases, and the N6 chains in LP-ZnN3 polymerize into infinite nitrogen chains in HP-ZnN3 at 72 GPa, showing a narrow-band-gap-semiconductor to metallic phase transition. Interestingly, P-1-ZnN3 has a superconducting transition temperature of 6.2 K at 50 GPa and 16.3 K at 100 GPa. In Cmm2-ZnN5 and Pcc2-ZnN6, nitrogen atoms polymerize into three-dimensional network structures and network layers under high pressures. Those predicted structures may enrich the phase diagram of high-pressure zinc nitrides, and provide clues for synthesis and exploration of novel stable polymeric nitrogen.