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One of the characteristics of topological materials is their non-trivial Berry phase. Experimental determination of this phase largely relies on a phase analysis of quantum oscillations. We study the angular dependence of the oscillations in a Dirac material ZrTe₅ and observe a striking spin zero effect, i.e., vanishing oscillations accompanied with a phase inversion. This indicates that the Berry phase in ZrTe₅ remains non-trivial for arbitrary field direction, in contrast with previous reports. The Zeeman splitting is found to be proportional to the magnetic field based on the condition for the spin zero effect in a Dirac band. Moreover, it is suggested that the Dirac band in ZrTe₅ is likely transformed into a line-node other than Weyl points for the field directions at which the spin zero occurs. The results underline a largely overlooked spin factor when determining the Berry phase from quantum oscillations.