  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.
