  We study a set of topological roots of the local Bernstein-Sato polynomial of
arbitrary plane curve singularities. These roots are characterized in terms of
certain divisorial valuations and the numerical data of the minimal log
resolution. In particular, this set of roots strictly contains both the
opposites of the jumping numbers in $(0, 1)$ and the poles of the motivic zeta
function counted with multiplicity. As a consequence, we prove the multiplicity
part of the Strong Monodromy Conjecture for $n = 2$.
