  In the process of studying the zeta-function for one parameter families of
Calabi-Yau manifolds we have been led to a manifold, first studied by Verrill,
for which the quartic numerator of the zeta-function factorises into two
quadrics remarkably often. Among these factorisations, we find persistent
factorisations; these are determined by a parameter that satisfies an algebraic
equation with coefficients in Q, so independent of any particular prime. Such
factorisations are expected to be modular with each quadratic factor associated
to a modular form. If the parameter is defined over Q this modularity is
assured by the proof of the Serre Conjecture. We identify three values of the
parameter that give rise to persistent factorisations, one of which is defined
over Q, and identify, for all three cases, the associated modular groups. We
note that these factorisations are due a splitting of Hodge structure and that
these special values of the parameter are rank two attractor points in the
sense of IIB supergravity. To our knowledge, these points provide the first
explicit examples of non-singular, non-rigid rank two attractor points for
Calabi-Yau manifolds of full SU(3) holonomy. The values of the periods and
their covariant derivatives, at the attractor points, are identified in terms
of critical values of the L-functions of the modular groups. Thus the critical
L-values enter into the calculation of physical quantities such as the area of
the black hole in the 4D spacetime. In our search for additional rank two
attractor points, we perform a statistical analysis of the numerator of the
zeta-function and are led to conjecture that the coefficients in this
polynomial are distributed according to the statistics of random USp(4)
matrices.
