Functional equation of an L-function (reviewed)

All known analytic L-functions have a functional equation that can be written in the form \[ \Lambda(s) := N^{s/2} \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), \] where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer or half-integer, \[ \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, \] and $\varepsilon$ is the sign of the functional equation. With those restrictions on the spectral parameters, the data in the functional equation is specified uniquely. The integer $d = J + 2 K$ is the degree of the L-function. The integer $N$ is the conductor (or level) of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters in the functional equation can be used to make up the 4-tuple called the Selberg data.

The axioms of the Selberg class are less restrictive than given above.

Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$.

For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers.