On the mean value of $\mathrm{GL}_1$ and $\mathrm{GL}_2$ $L$-functions, with applications to murmurations

We determine the mean value of $L$-functions attached to quadratic twists of automorphic representations on $\mathrm{GL}_1$ or $\mathrm{GL}_2$ in large regions of the critical strip. In the case of $\mathrm{GL}_1$, we go on to exhibit a recently discovered type of fine structure called "murmurations" unconditionally for all of our families. Our main tool is a new variant of the approximate functional equation imbued with a mechanism for dynamically rebalancing error terms while preserving holomorphicity.