For arbitrary Fermionic self energy, \Sigma(i wn) with wn=(2n+1)pi T, its real part is always an even function of wn while its imaginary part is always an odd function of wn.
Sorry, I think the issues about this analytic property of self energy is well known, or easily derived. I was curious why people would not enforce it in their calculation when say they design self energy ansatz for possible analytic continuation.