# Properties

 Label 2-101430-1.1-c1-0-80 Degree $2$ Conductor $101430$ Sign $1$ Analytic cond. $809.922$ Root an. cond. $28.4591$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 2-s + 4-s − 5-s + 8-s − 10-s + 2·11-s + 6·13-s + 16-s + 6·17-s − 20-s + 2·22-s + 23-s + 25-s + 6·26-s − 4·29-s + 2·31-s + 32-s + 6·34-s + 4·37-s − 40-s + 2·41-s − 4·43-s + 2·44-s + 46-s + 50-s + 6·52-s + 6·53-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 0.603·11-s + 1.66·13-s + 1/4·16-s + 1.45·17-s − 0.223·20-s + 0.426·22-s + 0.208·23-s + 1/5·25-s + 1.17·26-s − 0.742·29-s + 0.359·31-s + 0.176·32-s + 1.02·34-s + 0.657·37-s − 0.158·40-s + 0.312·41-s − 0.609·43-s + 0.301·44-s + 0.147·46-s + 0.141·50-s + 0.832·52-s + 0.824·53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 101430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 101430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$101430$$    =    $$2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 23$$ Sign: $1$ Analytic conductor: $$809.922$$ Root analytic conductor: $$28.4591$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{101430} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 101430,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$5.774899577$$ $$L(\frac12)$$ $$\approx$$ $$5.774899577$$ $$L(\frac{3}{2})$$ 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$$
5 $$1 + T$$
7 $$1$$
23 $$1 - T$$
good11 $$1 - 2 T + p T^{2}$$
13 $$1 - 6 T + p T^{2}$$
17 $$1 - 6 T + p T^{2}$$
19 $$1 + p T^{2}$$
29 $$1 + 4 T + p T^{2}$$
31 $$1 - 2 T + p T^{2}$$
37 $$1 - 4 T + p T^{2}$$
41 $$1 - 2 T + p T^{2}$$
43 $$1 + 4 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 - 4 T + p T^{2}$$
61 $$1 + p T^{2}$$
67 $$1 - 12 T + p T^{2}$$
71 $$1 - 8 T + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 + 8 T + p T^{2}$$
83 $$1 - 10 T + p T^{2}$$
89 $$1 + 2 T + p T^{2}$$
97 $$1 - 2 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$