$\def\A{\mathbb{A}} \def\B{\mathbb{B}} \def\C{\mathbb{C}} \newcommand{\Cx}{\mathbb{C}^{\times}} \def\F{\mathbb{F}} \def\G{\mathbb{G}} \def\H{\mathbb{H}} \def\K{\mathbb{K}} \def\M{\mathbb{M}} \def\N{\mathbb{N}} \def\O{\mathcal{O}} \def\P{\mathbb{P}} \def\Q{\mathbb{Q}} \def\R{\mathbb{R}} \def\T{\mathbb{T}} \def\V{\mathbb{V}} \def\Z{\mathbb{Z}} \def\fin{\mathrm{fin}} \def\x{^{\times}} \DeclareMathOperator{\Tr}{Tr}$
Let $F$ be a number field with degree $n$. Let $r_1$ be the number of real primes and $r_2$ be the number of complex primes. For a non-zero integral ideal $\mathfrak{m}$ of $F$, we put $ I_F(\mathfrak{m})=\{a \in I_F \mid(a, \mathfrak{m})=1\} $ where $I_F$ be the group of fractional ideals of $F$.
First we would like to recall the definition of a Grössencharacter.
Let $\mathcal{O}_F(\mathfrak{m})$ be the integers of $F$ that are prime to $\mathfrak{m}$.
Definition. A group homomorphism $\psi:I_F(\mathfrak{m})\to\C\x$ is called the (Hecke) Grössencharacter of $\bmod{\mathfrak{m}}$ if it satisfies the following condition:
There exist a character $$\psi_{\fin}:(\mathcal{O}_F/\mathfrak{m})\x\to\C\x$$ and complex numbers $u_v, w_v\left(1 \leqq v \leqq r_1+r_2\right)$ such that $u_v \in \begin{cases}\{0,1\} & \left(v \leqq r_1\right) \\ \mathbb{Z} & \left(r_1+1 \leqq v\right)\end{cases}$, so that
$$\psi((\alpha))=\psi_{\fin}(\alpha)\psi_{\infty}(\alpha)\ (\alpha\in\mathcal{O}_F(\mathfrak{m})).$$
Where $$ \psi_{\infty}(a):=\prod_{v=1}^{r_1+r_2}\left(a_v /\left|a_v\right|\right)^{u_v}\left|a_v\right|^{w_v} $$
$\psi_{\fin}$ and $\psi_{\infty}$ are called the finite and infinite parts of $\psi$, respectively. $u_v, w_v\left(1 \leqq v \leqq r_1+r_2\right)$ are called infinite type of $\psi$.
Now let $\psi$ be a unitary primitive Grössencharacter of conductor $\mathfrak{f}$ with infinite type $u_v, iv_v\left(v_v\in\R, 1 \leqq v \leqq r_1+r_2\right)$ and $v_1+\cdots +v_{r_1+r_2}=0$ (I have been confusing since some auther don't require this condition). The following are quoted from Miyake "Modular Forms" p.90.
Let $\mathfrak{c}$ be an integral ideal of $F$ such that $\mathfrak{d}_F\mathfrak{f}\mathfrak{c}$ ($\mathfrak{d}_F$ be the different ideal of $F$) is principal and $(\mathfrak{f}, \mathfrak{c})=1$. Take an element $b$ of $\mathcal{O}_F$ so that $\mathfrak{d}_F\mathfrak{f}\mathfrak{c}=(b)$, and define the Gauss sum $G(\psi)$ of $\psi$ by $$ G(\psi)=\frac{\psi_{\infty}(b)}{\psi(\mathfrak{c})} \sum_{[a]\in \mathfrak{c} / \mathfrak{m}\mathfrak{c}} \psi_{\fin}(a) e^{2 \pi i \Tr_{F/\Q}(a / b)}. $$ The value $G(\psi)$ is independent of the choice of $\mathfrak{c}, b$ and a set of representatives. We put $$ \Lambda(s, \psi)=\left(\frac{2^{r_1}\left|D_F\right| N(\mathfrak{f})}{(2 \pi)^n}\right)^{s / 2} \prod_{v=1}^{r_1+r_2} \Gamma\left(\frac{n_v s+\left|u_v\right|+i v_v}{2}\right) L(s, \psi), $$ where $$ n_v= \begin{cases}1 & \left(v \leqq r_1\right) \\ 2 & \left(r_1+1 \leqq v\right) .\end{cases} $$
Now the functional equation for a Hecke $L$-function is as follows.
Theorem. Let $\psi$ be a primitive Grössencharacter of conductor $\mathfrak{f}$. $\Lambda(s, \psi)$ is analytically continued to a meromorphic function on the whole $s$-plane, and satisfies the functional equation where $$ \Lambda(1-s, \psi)=T(\psi) \Lambda(s, \bar{\psi}) $$ $$ \begin{aligned} T(\psi) & =\frac{2^{i (v_{r_1+1} +\cdots +v_{r_1+r_2})} G(\psi)}{i^{u_1+\cdots +u_{r_1+r_2}}\mathcal{N}(\mathfrak{f})^{1 / 2}}. \end{aligned} $$
Now in Neukirch "Algebraic number theory", it is formulated by using the notion of "ideal numbers". For this notion, I want you to see this question which is perhaps useful information.
First $m,d$ be the ideal number defined by the following:
Then the functional equation is stated as follows:
Question1. Please show me the proof of the each formulation of Miyake and Neukirch are equivalent.
Question2. Please show me the proof of the functional equation formulated by ideal (not Tate's thesis).
