# Functional equation of an $L$-function

The Riemann $\zeta$-function has functional equation

 (1)

Dirichlet $L$-functions satisfy the functional equation

 (2)

where if is even and if is odd, and .

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

 (3)

Here and are the number of real and complex conjugate embeddings , is the discriminant, and is the degree of .

$L$-functions associated with a newform satisfy the functional equation

 (4)

where if is even and if is odd.

$L$-functions associated with a Maass newform with eigenvalue on satisfy the functional equation

 (5)

where if is even and if is odd.

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

[This section needs a bit of work]

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