# Properties

 Degree 2 Conductor 1 Sign $1$ Self-dual yes Motivic weight 11

# Origins

(not yet available)

## Dirichlet series

 $L(s,f)$  = 1 − 0.530·2-s + 0.598·3-s − 0.718·4-s + 0.691·5-s − 0.317·6-s − 0.376·7-s + 0.911·8-s − 0.641·9-s − 0.366·10-s + 1.000·11-s − 0.430·12-s − 0.431·13-s + 0.199·14-s + 0.413·15-s + 0.235·16-s − 1.179·17-s + 0.340·18-s + 0.987·19-s − 0.496·20-s − 0.225·21-s − 0.530·22-s + 0.603·23-s + 0.545·24-s − 0.522·25-s + 0.228·26-s − 0.982·27-s + 0.270·28-s + ⋯

## Functional equation

\begin{align} \Lambda(s,f)=\mathstrut &\Gamma_{\C}(s+5.5) \cdot L(s,f)\cr =\mathstrut & \Lambda(1-s,f) \end{align}

## Invariants

 $d$ = $2$ $N$ = $1$    =    $1$ $\varepsilon$ = $1$ primitive : yes self-dual : yes Selberg data = $(2,\ 1,\ (\ :11/2),\ 1)$

## Euler product

$$$L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1}$$$

## Particular Values

$L(1/2,f) \approx 0.7921228386$ $L(1,f) \approx 0.839345512$