Eh, the example I had in mind when I was writing that comment was of the evaluations of the Riemann zeta function, where the actual "factor on the pi" is not all that important because there's other things being mixed in anyways.
[1] https://en.wikipedia.org/wiki/Particular_values_of_the_Riema...
- e^(iτ) = 1, meaning that taking a full revolution around the complex plane brings you back to 1, where you started
- e^(iτ/2) = -1, meaning that taking a half revolution takes you to the opposite of where you started, -1.
If you insist on the "contains the additive identity" formulation (which IMO is gratuitous because e^(ix) is all about the complex plane, not regular addition and multiplication), you can do either e^(iτ) + 0 = 1, or e^(iτ) - 1 = 0.