# Special values of Hecke $L$-function on imaginary quadratic fields

Let $$K$$ be an imaginary quadratic number field and $$\mathcal{O}_K$$ its ring of integers. Let $$\chi$$ be an algebraic Hecke character on $$K$$ with conductor $$\mathfrak{f}$$ and infinity type $$(a,b)$$, i.e.
$$\chi (\mathfrak{a}) = \epsilon(\alpha)\chi_\infty^{-1}(\alpha) = \epsilon(\alpha) \cdot \alpha^a \overline{\alpha}^b$$ where $$\mathfrak{a}=(\alpha)$$ for all $$\alpha \in K^\times$$ and $$(\mathfrak{a},\mathfrak{f})=1$$ and a finite order character $$\epsilon : (\mathcal{O}_K/\mathfrak{f})^\times \longrightarrow \mathbb{S}^1$$ One has an associated Hecke $$L$$-function $$\begin{equation} L(s,\chi) = \sum\limits_{\substack{0 \neq \mathfrak{a} \lhd \mathcal{O}_K \\ (\mathfrak{a},\mathfrak{f})=1 }} \frac{\chi(\mathfrak{a})}{N(\mathfrak{a})^s} \end{equation}$$ Which is absolutely convergent on $$\lbrace z \in \mathbb{C} \, | \, \operatorname{Re}(s) > \frac{a+b}{2}+1 \rbrace$$. Let $$P_\mathfrak{f}:= \lbrace \mathfrak{a}=(\alpha) \text{ principal fractional ideals } \: | \: \alpha \equiv 1 \: \operatorname{mod}^* \: \mathfrak{f} \rbrace$$ the subgroup of $$I(\mathfrak{f}):= \lbrace \mathfrak{a} \text{ fractional ideals of } K \: | \: (\mathfrak{a},\mathfrak{f})=1 \rbrace$$.

I am reading a paper and the author writes for $$a \in \mathbb{N}$$, $$s > \frac{a}{2}+1$$

\begin{align} L(s,\overline{\chi}^a) = \sum\limits_{\substack{0 \neq \mathfrak{a} \lhd \mathcal{O}_K \\ (\mathfrak{a},\mathfrak{f})=1 }} \frac{\overline{\chi}^a(\mathfrak{a})}{N(\mathfrak{a})^s} \underset{(1)}{=}& \frac{1}{\omega_\mathfrak{f}} \sum\limits_{\mathfrak{a} \in I(\mathfrak{f})/P_\mathfrak{f}} \frac{\overline{\chi}^a(\mathfrak{a})}{N(\mathfrak{a})^s} \sum\limits_{\substack{\alpha \in \mathfrak{a}^{-1} \\ \alpha \equiv 1 \: \operatorname{mod}^* \: \mathfrak{f} }}\frac{\overline{\chi}^a(\mathfrak{a})}{|\alpha|^{2s}} \\ \underset{(2)}{=}& \frac{1}{\omega_\mathfrak{f}} \sum\limits_{\mathfrak{a} \in I(\mathfrak{f})/P_\mathfrak{f}} \; \sum\limits_{\gamma \in \mathfrak{a}^{-1}\mathfrak{f}}\frac{(\overline{\chi(\alpha_\mathfrak{a} \mathfrak{a}) + \chi(\mathfrak{a})\gamma})^a}{|\chi(\alpha_\mathfrak{a} \mathfrak{a}) + \chi(\mathfrak{a})\gamma|^{2s}} \end{align}

Here he says that $$\omega_\mathfrak{f}$$ is the number of roots of unity in $$K$$ that are congruent to $$1$$ modulo $$\mathfrak{f}$$.

I don't understand from where it comes from, nor how he gets the equalities (1) and (2)... If anyone can help explaining, it would be very much appreciated.

P.S: Why does one have $$|\chi(\mathfrak{a})|^2=N(\mathfrak{a})$$ in (1) ?

When $$\psi$$ is a character of $$C_K$$ then $$L(s,\psi ) =\sum_{0\ne I\subset O_K} \psi(I)N(I)^{-s}= \sum_{c\in C_K} \psi(c) \sum_{I\subset O_K,I\sim c} N(I)^{-s}$$ $$= \sum_{c\in C_K} \psi(c) \sum_{b\in (J_c-0)^{-1}/O_K^\times} N(J_c b)^{-s} = \sum_{c\in C_K} \psi(c)N(J_c)^{-s} \sum_{b\in (J_c-0)^{-1}/O_K^\times} |N_{K/Q}( b)|^{-s}$$
where $$J_c$$ is any fixed ideal in the class $$c$$.
(this is because $$I$$ is in the same class as $$J_c$$ iff $$I = b J_c$$ with $$0\ne b\in J_c^{-1} = \{ d\in K, dJ_c\subset O_K\}$$)
In your question it works the same way except that $$\psi$$ is a character of the class group of an order multiplied by a character of infinite place, and $$O_K^\times$$ is finite because $$K$$ is an imaginary quadratic field.
The point of doing so is that $$\sum_{b\in (J_c-0)^{-1}/O_K^\times} |N_{K/Q}( b)|^{-s}$$ is the Mellin transform of some kind of $$\theta$$ function.