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

We generalize our previous method on subconvexity problem for $\mathrm{GL}_2 \times \mathrm{GL}_1$ with cuspidal representations to Eisenstein series, and deduce 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 regularized integral due to Zagier developed in a previous paper to obtain the relevant triple product formulas of Eisenstein series.