  The partition function of 2d conformal field theory is a modular invariant
function. It is known that the partition function of a holomorphic CFT whose
central charge is a multiple of 24 is a polynomial in the Klein function. In
this paper, by using the medium temperature expansion we show that every
modular invariant partition function can be mapped to a holomorphic partition
function whose structure can be determined similarly. We use this map to study
partition function of CFTs with half-integer left and right conformal weights.
We show that the corresponding left and right central charges are necessarily
multiples of 4. Furthermore, the degree of degeneracy of high-energy levels can
be uniquely determined in terms of the degeneracy in the low energy states.
