  Let $E$ be an elliptic curve over $\mathbf{Q}$. We conjecture asymptotic
estimates for the number of vanishings of $L(E,1,\chi)$ as $\chi$ varies over
all primitive Dirichlet characters of orders 4 and 6, subject to a mild
hypothesis on $E$. Our conjectures about these families come from conjectures
about random unitary matrices as predicted by the philosophy of Katz-Sarnak. We
support our conjectures with numerical evidence.
  Earlier work by David, Fearnley and Kisilevsky formulates analogous
conjectures for characters of any odd prime order. In the composite order case,
however, we need to justify our use of random matrix theory heuristics by
analyzing the equidistribution of the squares of normalized Gauss sums. Along
the way we introduce the notion of totally order $\ell$ characters to quantify
how quickly quartic and sextic Gauss sums become equidistributed. Surprisingly,
the rate of equidistribution in the full family of quartic (sextic, resp.)
characters is much slower than in the sub-family of totally quartic (sextic,
resp.) characters. A conceptual explanation for this phenomenon is that the
full family of order $\ell$ twisted elliptic curve $L$-functions, with $\ell$
even and composite, is a mixed family with both unitary and orthogonal aspects.
