# Properties

 Label 2-200-1.1-c1-0-1 Degree $2$ Conductor $200$ Sign $1$ Analytic cond. $1.59700$ Root an. cond. $1.26372$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 4·7-s − 3·9-s + 4·11-s + 2·13-s − 2·17-s + 4·19-s − 4·23-s − 2·29-s − 8·31-s − 6·37-s − 6·41-s + 8·43-s − 4·47-s + 9·49-s − 6·53-s − 4·59-s − 2·61-s − 12·63-s − 8·67-s + 6·73-s + 16·77-s + 9·81-s + 16·83-s − 6·89-s + 8·91-s + 14·97-s − 12·99-s + ⋯
 L(s)  = 1 + 1.51·7-s − 9-s + 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s − 0.371·29-s − 1.43·31-s − 0.986·37-s − 0.937·41-s + 1.21·43-s − 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.520·59-s − 0.256·61-s − 1.51·63-s − 0.977·67-s + 0.702·73-s + 1.82·77-s + 81-s + 1.75·83-s − 0.635·89-s + 0.838·91-s + 1.42·97-s − 1.20·99-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$200$$    =    $$2^{3} \cdot 5^{2}$$ Sign: $1$ Analytic conductor: $$1.59700$$ Root analytic conductor: $$1.26372$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{200} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 200,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.327698878$$ $$L(\frac12)$$ $$\approx$$ $$1.327698878$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3 $$1 + p T^{2}$$
7 $$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}$$
19 $$1 - 4 T + p T^{2}$$
23 $$1 + 4 T + p T^{2}$$
29 $$1 + 2 T + p T^{2}$$
31 $$1 + 8 T + p T^{2}$$
37 $$1 + 6 T + p T^{2}$$
41 $$1 + 6 T + p T^{2}$$
43 $$1 - 8 T + p T^{2}$$
47 $$1 + 4 T + p T^{2}$$
53 $$1 + 6 T + p T^{2}$$
59 $$1 + 4 T + p T^{2}$$
61 $$1 + 2 T + p T^{2}$$
67 $$1 + 8 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 + p T^{2}$$
83 $$1 - 16 T + p T^{2}$$
89 $$1 + 6 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}$$