For the driving point impedance N(s) (with common factors p(s) and q(s) canceled):
1
the poles and zeros of N(s) must be infinity if imaginary or complex
2
the poles and zeros of N(s) must be same if imaginary or complex.
3
the poles and zeros of N(s) must be complex conjugate if imaginary or complex
4
the poles and zeros of N(s) must be zero if imaginary and complex.