The value distribution of derivatives of characteristic polynomials of
matrices from SO(N) is calculated at the point 1, the symmetry point on the
unit circle of the eigenvalues of these matrices. We consider subsets of
matrices from SO(N) that are constrained to have $n$ eigenvalues equal to 1,
and investigate the first non-zero derivative of the characteristic polynomial
at that point.
The connection between the values of random matrix characteristic polynomials
and values of $L$-functions in families has been well-established. The
motivation for this work is the expectation that through this connection with
$L$-functions derived from families of elliptic curves, and using the Birch and
Swinnerton-Dyer conjecture to relate values of the $L$-functions to the rank of
elliptic curves, random matrix theory will be useful in probing important
questions concerning these ranks.