  We study the individual digits for the absolute value of the characteristic
polynomial for the Circular $\beta$-Ensemble. We show that, in the large $N$
limit, the first digits obey Benford's Law and the further digits become
uniformly distributed. Key to the proofs is a bound on the rate of convergence
in total variation norm in the CLT for the logarithm of the absolute value of
the characteristic polynomial.
