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

"Murmurations" are a recently-discovered type of fine structure in sums of Dirichlet coefficients averaged over families of $L$-functions. The root cause of this phenomenon remains mysterious. In the present paper, we demonstrate how murmurations arise from the averaging of approximate functional equations. This approach to the study of murmurations explains their empirically observed ubiquity, as well as their characteristic scale invariance and the peculiar normalization they demand. We implement our new approach to the study of murmurations in the case of quadratic twist families of $\mathrm{GL}_1$ automorphic representations, where we exhibit murmurations unconditionally. Our proof centres around estimating mean values of the $L$-functions in our quadratic twist families. In particular, we require estimates valid significantly higher in the critical strip than what existing results provide. To produce these estimates, we construct a variation of the approximate functional equation which is imbued with a mechanism for dynamically rebalancing error terms while preserving holomorphicity. We also generalize and sharpen results of Jutila and Stankus on sums of quadratic characters and fundamental discriminants. Mean value estimates are given for $\mathrm{GL}_2$ quadratic twist families as well.