# Properties

 Degree $2$ Conductor $1122$ Sign $1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s + 9-s + 11-s + 12-s − 4·13-s + 2·14-s + 16-s − 17-s + 18-s + 8·19-s + 2·21-s + 22-s + 6·23-s + 24-s − 5·25-s − 4·26-s + 27-s + 2·28-s + 6·29-s − 4·31-s + 32-s + 33-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 1.83·19-s + 0.436·21-s + 0.213·22-s + 1.25·23-s + 0.204·24-s − 25-s − 0.784·26-s + 0.192·27-s + 0.377·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.174·33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1122$$    =    $$2 \cdot 3 \cdot 11 \cdot 17$$ Sign: $1$ Motivic weight: $$1$$ Character: $\chi_{1122} (1, \cdot )$ Sato-Tate group: $\mathrm{SU}(2)$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1122,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$3.366050137$$ $$L(\frac12)$$ $$\approx$$ $$3.366050137$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - T$$
3 $$1 - T$$
11 $$1 - T$$
17 $$1 + T$$
good5 $$1 + p T^{2}$$
7 $$1 - 2 T + p T^{2}$$
13 $$1 + 4 T + p T^{2}$$
19 $$1 - 8 T + p T^{2}$$
23 $$1 - 6 T + p T^{2}$$
29 $$1 - 6 T + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
37 $$1 - 2 T + p T^{2}$$
41 $$1 + 6 T + p T^{2}$$
43 $$1 + 4 T + p T^{2}$$
47 $$1 - 6 T + p T^{2}$$
53 $$1 + 12 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 + 4 T + p T^{2}$$
67 $$1 + 4 T + p T^{2}$$
71 $$1 - 6 T + p T^{2}$$
73 $$1 - 2 T + p T^{2}$$
79 $$1 + 10 T + p T^{2}$$
83 $$1 - 12 T + p T^{2}$$
89 $$1 - 18 T + p T^{2}$$
97 $$1 - 14 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$