CUPID-0 is the first pilot experiment of CUPID, a next-generation project searching for neutrino-less double beta decay. In its first scientific run, CUPID-0 operated 26 ZnSe cryogenic calorimeters coupled to light detectors in the underground Laboratori Nazionali del Gran Sasso. In this work, we analyzed a ZnSe exposure of 11.34 kg$times$yr to search for the neutrino-less double beta decay of $^{70}$Zn and for the neutrino-less positron-emitting electron capture of $^{64}$Zn. We found no evidence for these decays and set 90$%$ credible interval limits of ${rm T}_{1/2}^{0 ubetabeta}(^{70}{rm Zn}) > 1.6 times 10^{21}$ yr and ${rm T}_{1/2}^{0 u EC beta+}(^{64}{rm Zn}) > 1.2 times 10^{22}$ yr, surpassing by almost two orders of magnitude the previous experimental results
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