The values for $\operatorname{SL}(2,q)$ immediately give the following bounds.

$d_{(q\pm 1)/2}^\natural \leq \frac{2}{q}$ for $q$ any odd prime power greater than 3.

Note the upper bound in the corollary for $q=7$ is the exact value of $d^\natural_3$.

Even though Theorem \[thm1\] implies $d_n^\natural$ is not a decreasing sequence for $n \ge 1$, this corollary at least suggests that $d_n^\natural \to 0$ as $n \to \infty$.

The proof of Theorem \[thm1\] relies on consideration of various cases according to the possible finite primitive subgroups of $\operatorname{GL}_2({\mathbb C})$ and $\operatorname{GL}_3({\mathbb C})$ which are “minimal lifts”, and about half of these are of the form $\operatorname{PSL}(2,q)$ or $\operatorname{SL}(2,q)$ for $q \in \{ 3, 5, 7, 9 \}$. Thus Theorem \[thm2\] is a generalization of one of the ingredients for Theorem \[thm1\]. However, most of the work involved in the proof of Theorem \[thm1\] is the determination of and reduction to these minimal lifts, as described in Section \[sec:general\].

Distinguishing $L$-functions
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Let $F$ be a number field, and consider an $L$-function $L(s)$, which is a meromorphic function of a complex variable $s$ satisfying certain properties, principally having an Euler product $L(s) = \prod L_v(s)$ where $v$ runs over all primes of $F$ for 