The Chebotarev density theorem roughly states that the number of unramified primes $\mathfrak{p}$ of $L/K$ with conjugacy class $\sigma_{\mathfrak{p}}=\mathcal{C}$ with norm less than $x$, $\pi_{\mathcal{C}}(x)$, satisfies
$\pi_{\mathcal{C}}(x)\sim|\mathcal{C}|/|\mathcal{G}|Li(x)$,
where $|\mathcal{G}|$ is the size of the Galois group of $L/K$. Very roughly speaking, there are more unramified primes $\mathfrak{p}$ of conjugacy class $\mathcal{C}$ less than a certain norm if the size of the conjugacy class is larger.
However, in most bounds for the first (smallest norm) unramified prime $\mathfrak{p}$ of conjugacy class $\mathcal{C}$, the size of the particular conjugacy class is not considered:
With $N(\mathcal{C})$ the smallest norm of an unramified prime $\mathfrak{p}$ with conjugacy class $\mathcal{C}$ and $d_L$ the size of the discriminant of the field extension:
$N(\mathcal{C})\leq kd_L^{16}$ for sufficiently large $d_L$,
$N(\mathcal{C})\leq kd_L^{12577}$ unconditionally.*
The size of the conjugacy class is NOT considered whatsover, even though one many intuit that the bound should inversely depend on the size of $\mathcal{C}$. This leads to my question:
Is there a known bound on smallest unramified primes of a certain conjugacy class that depends on the size of $\mathcal{C}$?