 Covolume-one theta function of a fractional ideal

Let $C=\sqrt{|D_K|}N(\mathfrak{b})$ and $n=[K:\mathbb{Q}]$. Normalize the Minkowski embedding of a number field to the covolume-one Euclidean lattice ${\Lambda }_{\mathfrak{b}}=C^{-1/n}j(\mathfrak{b})$, using the metric with complex coordinates weighted by $2$. Its theta function isWriting $\Vert y\Vert ={\prod }_v{y}_v^{[K_v:\mathbb{R}]}$ and using the trace dual of a fractional ideal gives $\Theta (y,\mathfrak{b})=\Vert y{\Vert }^{-1/2}\Theta (y^{-1},{\mathfrak{b}}^{\vee })$. Its small-parameter asymptotic of a lattice theta sum has leading coefficient one. For the unscaled ideal lattice, the leading coefficient is instead $C^{-1}$; one must specify the normalization when quoting this limit.