  In this paper we obtain a precise formula for the $1$-level density of
$L$-functions attached to non-Galois cubic Dedekind zeta functions. We find a
secondary term which is unique to this context, in the sense that no
lower-order term of this shape has appeared in previously studied families. The
presence of this new term allows us to deduce an omega result for cubic field
counting functions, under the assumption of the Generalized Riemann Hypothesis.
We also investigate the associated $L$-functions Ratios Conjecture, and find
that it does not predict this new lower-order term. Taking into account the
secondary term in Roberts' Conjecture, we refine the Ratios Conjecture to one
which captures this new term. Finally, we show that any improvement in the
exponent of the error term of the recent Bhargava--Taniguchi--Thorne cubic
field counting estimate would imply that the best possible error term in the
refined Ratios Conjecture is $O_\varepsilon(X^{-\frac 13+\varepsilon})$. This
is in opposition with all previously studied families, in which the expected
error in the Ratios Conjecture prediction for the $1$-level density is
$O_\varepsilon(X^{-\frac 12+\varepsilon})$.
