A filter has a zero-phase response if its frequency response is real-valued
A zero-phase filter must have an odd-number of samples in its support with the origin at the center.
If the filter has real coefficients, then the impulse response must be symmetric about the origin, i.e. h(n) = h(-n). If the filter has imaginary coefficients, then the impulse response must be anti-symmetric about the origin, i.e. h(n) = -h(-n).