Goldfeld proved the following result. Let $E$ be an elliptic curve (with conductor $N$) over $\mathbb{Q}$ whose Hasse-Weil L-function has a zero at $s = 1$ with multiplicity $g$ then for sufficiently large $D > 0$ then the class number $h(D)$ of an imaginary quadratic field with discriminant $-D$ satisfies the following inequality if $(N, D) = 1$, $$h(D) > c(g, N) {\log(D)}^{g - \mu - 1} \exp(-21\sqrt{g\log(\log(D))})$$ where $\mu = 1, 2$ depending on $\chi_{D}(-N) = {(-1)}^{g - \mu}$, $\chi_{D}$ being the quadratic character with conductor $-D$.
Now, Goldfeld claims that if there is an elliptic curve with $g = 3$ (which later Gross - Zagier came up with), then we can effectively find bounds for $D$ that satisfy $h(D) = n$ for any fixed $n$. What I am unable to understand is the fact that suppose with $g = 3$ it happened to be that $\mu = 2$, the above stated lower bound on $h(D)$ would we decreasing with $D$ and we wouldn't be able to bound $D$ (from above). For a fixed elliptic curve $\mu = \mu(D)$ is a function of $D$. So my question is, in that case how is it possible to conclude anything useful when $\mu = 2$ i.e. when $\chi_D(-N) = -1$ with $g = 3$.
