# Properties

 Degree 2 Conductor $2 \cdot 3$ Sign $1$ Motivic weight 3 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2·2-s − 3·3-s + 4·4-s + 6·5-s + 6·6-s − 16·7-s − 8·8-s + 9·9-s − 12·10-s + 12·11-s − 12·12-s + 38·13-s + 32·14-s − 18·15-s + 16·16-s − 126·17-s − 18·18-s + 20·19-s + 24·20-s + 48·21-s − 24·22-s + 168·23-s + 24·24-s − 89·25-s − 76·26-s − 27·27-s − 64·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.536·5-s + 0.408·6-s − 0.863·7-s − 0.353·8-s + 1/3·9-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 + 1/4·16-s − 1.79·17-s − 0.235·18-s + 0.241·19-s + 0.268·20-s + 0.498·21-s − 0.232·22-s + 1.52·23-s + 0.204·24-s − 0.711·25-s − 0.573·26-s − 0.192·27-s − 0.431·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$6$$    =    $$2 \cdot 3$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : $\chi_{6} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 6,\ (\ :3/2),\ 1)$ $L(2)$ $\approx$ $0.509710$ $L(\frac12)$ $\approx$ $0.509710$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + p T$$
3 $$1 + p T$$
good5 $$1 - 6 T + p^{3} T^{2}$$
7 $$1 + 16 T + p^{3} T^{2}$$
11 $$1 - 12 T + p^{3} T^{2}$$
13 $$1 - 38 T + p^{3} T^{2}$$
17 $$1 + 126 T + p^{3} T^{2}$$
19 $$1 - 20 T + p^{3} T^{2}$$
23 $$1 - 168 T + p^{3} T^{2}$$
29 $$1 - 30 T + p^{3} T^{2}$$
31 $$1 + 88 T + p^{3} T^{2}$$
37 $$1 - 254 T + p^{3} T^{2}$$
41 $$1 - 42 T + p^{3} T^{2}$$
43 $$1 + 52 T + p^{3} T^{2}$$
47 $$1 + 96 T + p^{3} T^{2}$$
53 $$1 - 198 T + p^{3} T^{2}$$
59 $$1 + 660 T + p^{3} T^{2}$$
61 $$1 + 538 T + p^{3} T^{2}$$
67 $$1 - 884 T + p^{3} T^{2}$$
71 $$1 - 792 T + p^{3} T^{2}$$
73 $$1 - 218 T + p^{3} T^{2}$$
79 $$1 + 520 T + p^{3} T^{2}$$
83 $$1 + 492 T + p^{3} T^{2}$$
89 $$1 - 810 T + p^{3} T^{2}$$
97 $$1 - 1154 T + p^{3} T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}