  The hot nucleus $^{162}\mathrm{Dy}$ is investigated using covariant density
functional theory, where the shell-model-like approach treats the pairing
correlation. Lee-Yang's theorem is applied to classify the pairing phase
transition by analyzing the distribution of zeros of the partition function in
the complex temperature plane. The distribution of zeros of the partition
function converges with increasing particle numbers and illustrates the
characteristics of the phase transition. In our calculations, we determine the
first order of the phase transition near the critical temperature. Different
seniority states show the pairing phase transition from a superfluid to a
normal phase, ranging from fully paired states to completely unpaired states.
