  On the critical line the conditional distribution of the zeta function's
magnitude around zeta zeros exists and predicts the well-known pair correlation
between nontrivial zeta zeros. However, this conditional distribution does not
exist at most distances above or below any nontrivial zeta zeros that are off
the critical line. This shows that the zeta function's magnitude cannot have
vertical statistical structure at most distances around nontrivial zeta zeros
off the critical line. The proofs of these results are straightforward, using
only statistical properties of certain prime sums, elementary properties of
normal and elliptical random variables, and the pole structure of the zeta
function. These results readily generalize to L-functions.
