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We report on the measurement of new low-lying states in the neutron-rich 81,82,83,84 Zn nuclei via in-beam γ -ray spectroscopy. These include the 41+→21+ transition in 82 Zn, the 21+→0g.s.+ and 41+→21+ transitions in 84 Zn, and low-lying states in 81,83 Zn were observed for the first time. The reduced E(21+) energies and increased E(41+)/E(2+1) ratios at N=52, 54 compared to those in 80 Zn attest that the magicity is confined to the neutron number N=50 only. The deduced level schemes are compared to three state-of-the-art shell model calculations and a good agreement is observed with all three calculations. The newly observed 2+ and 4+ levels in 84 Zn suggest the onset of deformation towards heavier Zn isotopes, which has been incorporated by taking into account the upper sdg orbitals in the Ni78-II and the PFSDG-U models.