Root numbers

The sign of an L-function is the complex number $\varepsilon$ occurring in its functional equation

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

It appears as the fourth component of the Selberg data of L.

If all of the coefficients of the Dirichlet series defining $L(s)$ are real, then necessarily $\varepsilon = \pm 1$. If the coefficients are real and $\varepsilon = - 1$, then $L(1/2)=0$.