  Many topological or critical aspects of the Kitaev chain are well known, with
several classic results. In contrast, the study of the critical behavior of the
strong Majorana zero modes (MZM) has been overlooked. Here we introduce two
topological markers which, surprisingly, exhibit non-trivial signatures over
the entire (1+1) Ising critical line. We first analytically compute the MZM
fidelity ${\cal{F}}_{\rm MZM}$--a measure of the MZM mapping between parity
sectors. It takes a universal value along the (1+1) Ising critical line,
${\cal{F}}_{\rm MZM}=\sqrt{8}/\pi$, independent of the energy. We also obtain
an exact analytical result for the critical MZM occupation number ${{\cal
N}}_{\rm MZM}$ which depends on the Catalan's constant ${\cal G}\approx
0.91596559$, for both the ground-state (${{\cal N}}_{\rm
MZM}=1/2-4{\cal{G}}/\pi^2\approx 0.12877$) and the first excited state (${{\cal
N}}_{\rm MZM}=1/2+(8-4{\cal{G}})/\pi^2\approx 0.93934$). We further compute
finite-size corrections which identically vanish for the special ratio
$\Delta/t=\sqrt{2}-1$ between pairing and hopping in the critical Kitaev chain.
