# Properties

 Label 2-20e2-4.3-c0-0-0 Degree $2$ Conductor $400$ Sign $1$ Analytic cond. $0.199626$ Root an. cond. $0.446795$ Motivic weight $0$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 9-s − 2·29-s − 2·41-s + 49-s − 2·61-s + 81-s − 2·89-s + 2·101-s + 2·109-s + ⋯
 L(s)  = 1 + 9-s − 2·29-s − 2·41-s + 49-s − 2·61-s + 81-s − 2·89-s + 2·101-s + 2·109-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$400$$    =    $$2^{4} \cdot 5^{2}$$ Sign: $1$ Analytic conductor: $$0.199626$$ Root analytic conductor: $$0.446795$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: $\chi_{400} (351, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 400,\ (\ :0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8675353414$$ $$L(\frac12)$$ $$\approx$$ $$0.8675353414$$ $$L(1)$$ 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 - T )( 1 + T )$$
7 $$( 1 - T )( 1 + T )$$
11 $$( 1 - T )( 1 + T )$$
13 $$1 + T^{2}$$
17 $$1 + T^{2}$$
19 $$( 1 - T )( 1 + T )$$
23 $$( 1 - T )( 1 + T )$$
29 $$( 1 + T )^{2}$$
31 $$( 1 - T )( 1 + T )$$
37 $$1 + T^{2}$$
41 $$( 1 + T )^{2}$$
43 $$( 1 - T )( 1 + T )$$
47 $$( 1 - T )( 1 + T )$$
53 $$1 + T^{2}$$
59 $$( 1 - T )( 1 + T )$$
61 $$( 1 + T )^{2}$$
67 $$( 1 - T )( 1 + T )$$
71 $$( 1 - T )( 1 + T )$$
73 $$1 + T^{2}$$
79 $$( 1 - T )( 1 + T )$$
83 $$( 1 - T )( 1 + T )$$
89 $$( 1 + T )^{2}$$
97 $$1 + T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$