  We make the polynomial dependence on the fixed representation $\pi$ in our
previous subconvex bound of $L(1/2,\pi \otimes \chi)$ for $\mathrm{GL}_2 \times
\mathrm{GL}_1$ explicit, especially with respect to the usual conductor
$\mathbf{C}(\pi_{\mathrm{fin}})$. There is no clue that the original choice,
due to Michel & Venkatesh, of the test function at the infinite places should
be the optimal one. Hence we also investigate a possible variant of such local
choices in some special situations.

Number Theory