Conductor of an L-function (reviewed)

The conductor of an L-function is the integer $N$ occurring in its functional equation

\[ \Lambda(s) := N^{s/2} \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). \]

The conductor of an analytic L-function is the second component in the Selberg data. For a Dirichlet L-function associated with a primitive Dirichlet character, the conductor of the L-function is the same as the conductor of the character. For a primitive L-function associated with a cusp form $\phi$ on $GL(2)/\mathbb Q$, the conductor of the L-function is the same as the level of $\phi$.

In the literature, the word level is sometimes used instead of conductor.