  The functional determinant of a special second order quantum-mechanical
operator is calculated with its zero mode gauged out by the method of the
Faddeev-Popov gauge fixing procedure. This operator subject to periodic
boundary conditions arises in applications of the early Universe theory and, in
particular, determines the one-loop statistical sum in quantum cosmology
generated by a conformal field theory (CFT). The calculation is done for a
special case of a periodic zero mode of this operator having two roots (nodes)
within the period range, which corresponds to the class of cosmological
instantons in the CFT driven cosmology with one oscillation of the cosmological
scale factor of its Euclidean Friedmann-Robertson-Walker metric.
