# Properties

 Degree $2$ Conductor $13680$ Sign $1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $2$

# Related objects

## Dirichlet series

 L(s)  = 1 − 5-s − 4·7-s − 4·11-s − 2·13-s + 2·17-s + 19-s − 8·23-s + 25-s − 6·29-s − 4·31-s + 4·35-s − 10·37-s + 2·41-s − 12·43-s + 9·49-s − 6·53-s + 4·55-s − 10·61-s + 2·65-s + 4·67-s − 8·71-s + 2·73-s + 16·77-s + 12·79-s − 8·83-s − 2·85-s − 6·89-s + ⋯
 L(s)  = 1 − 0.447·5-s − 1.51·7-s − 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.229·19-s − 1.66·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s + 0.676·35-s − 1.64·37-s + 0.312·41-s − 1.82·43-s + 9/7·49-s − 0.824·53-s + 0.539·55-s − 1.28·61-s + 0.248·65-s + 0.488·67-s − 0.949·71-s + 0.234·73-s + 1.82·77-s + 1.35·79-s − 0.878·83-s − 0.216·85-s − 0.635·89-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 13680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 13680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$13680$$    =    $$2^{4} \cdot 3^{2} \cdot 5 \cdot 19$$ Sign: $1$ Motivic weight: $$1$$ Character: $\chi_{13680} (1, \cdot )$ Sato-Tate group: $\mathrm{SU}(2)$ Primitive: yes Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(2,\ 13680,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1 + T$$
19 $$1 - T$$
good7 $$1 + 4 T + p T^{2}$$
11 $$1 + 4 T + p T^{2}$$
13 $$1 + 2 T + p T^{2}$$
17 $$1 - 2 T + p T^{2}$$
23 $$1 + 8 T + p T^{2}$$
29 $$1 + 6 T + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
37 $$1 + 10 T + p T^{2}$$
41 $$1 - 2 T + p T^{2}$$
43 $$1 + 12 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + 6 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 + 10 T + p T^{2}$$
67 $$1 - 4 T + p T^{2}$$
71 $$1 + 8 T + p T^{2}$$
73 $$1 - 2 T + p T^{2}$$
79 $$1 - 12 T + p T^{2}$$
83 $$1 + 8 T + p T^{2}$$
89 $$1 + 6 T + p T^{2}$$
97 $$1 - 18 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$