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We generalize our previous method on subconvexity problem for $\mathrm{GL}_2\times \mathrm{GL}_1$ with cuspidal representations to Eisenstein series, anddeduce a Burgess-like subconvex bound for Hecke characters, i.e., the bound$|L(1/2,\chi)| \ll_{\mathbf{F},\epsilon}\mathbf{C}(\chi)^{1/4-(1-2\theta)/16+\epsilon}$ for varying Hecke characters$\chi$ over a number field $\mathbf{F}$ with analytic conductor$\mathbf{C}(\chi)$. As a main tool, we apply the extended theory of regularizedintegral due to Zagier developed in a previous paper to obtain the relevanttriple product formulas of Eisenstein series.

Burgess-like subconvexity for $\mathrm{GL}_1$