Functional equation of an $L$-function

The Riemann $\zeta$-function has functional equation

$$\xi(s) = \pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s) \cr$$ (1)

Dirichlet $L$-functions satisfy the functional equation

$$\xi(s,\chi) = \pi^{-\frac{s}{2}}\Gamma(\frac{s}{2}+a)L(s,\chi) \cr$$ (2)

where $a=0$ if $\chi$ is even and $a=1$ if $\chi$ is odd, and $*****$.

The Dedekind zeta function of a number field $K$ satisfies the functional equation

$$\xi_K(s) = \left(\frac{\sqrt{\vert d_K\vert}}{2^{r_2}\pi^{n/2}}\right)^{s} \Gamma(s/2)^{r_1} \Gamma(s)^{r_2}\zeta_K(s)) \cr$$ (3)

Here $r_1$ and $2r_2$ are the number of real and complex conjugate embeddings $K\subset \mathbb C$, $d_K$ is the discriminant, and $n=[K,{\mathbb Q}]$ is the degree of $K/{\mathbb Q}$.

$L$-functions associated with a newform $f\in S_k(\Gamma_0(N)$ satisfy the functional equation

$$\xi(s,f) = \left(\frac{\pi}{N}\right)^{-s} \Gamma\left(s+\frac{k-1}{2}\right)L(s,f) \cr$$ (4)

where $a=0$ if $\chi$ is even and $a=1$ if $\chi$ is odd.

$L$-functions associated with a Maass newform with eigenvalue $\lambda=\frac14+R^2$ on $\Gamma_0(N)$ satisfy the functional equation

$$\xi(s,f) = \left(\frac{N}{\pi}\right)^{s} \Gamma\left(\frac{s+iR+a}{2}\right)\Gamma\left(\frac{s-iR+a}{2}\right) L(s,f) \cr$$ (5)

where $a=0$ if $f$ is even and $a=1$ if $f$ is odd.

$GL(r)$ $L$-functions satisfy functional equations of the form

$$\Phi(s):= \left(\frac{N}{\pi^r}\right)^{s/2} \prod_{j=1}^r \Gamma\left(\frac{s+r_j}{2}\right) F(s) = \varepsilon \Phi(1-s).$$

[This section needs a bit of work]

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