# Properties

 Degree 2 Conductor 6 Sign $1$ Self-dual yes Motivic weight 3

# Origins

(not yet available)

## Dirichlet series

 $L(s,f)$  = 1 − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.536·5-s + 0.408·6-s − 0.863·7-s − 0.353·8-s + 0.333·9-s − 0.379·10-s + 0.328·11-s − 0.288·12-s + 0.810·13-s + 0.610·14-s − 0.309·15-s + 0.250·16-s − 1.797·17-s − 0.235·18-s + 0.241·19-s + 0.268·20-s + 0.498·21-s − 0.232·22-s + 1.523·23-s + 0.204·24-s − 0.712·25-s − 0.573·26-s − 0.192·27-s − 0.431·28-s + ⋯

## Functional equation

\begin{align} \Lambda(s,f)=\mathstrut & 6 ^{s/2} \Gamma_{\C}(s+1.5) \cdot L(s,f)\cr =\mathstrut & \Lambda(1-s,f) \end{align}

## Invariants

 $d$ = $2$ $N$ = $6$    =    $2 \cdot 3$ $\varepsilon$ = $1$ primitive : yes self-dual : yes Selberg data = $(2,\ 6,\ (\ :3/2),\ 1)$

## 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

$L(1/2,f) \approx 0.5097104234$ $L(1,f) \approx 0.6311371888$