# Properties

 Label 2-8470-1.1-c1-0-196 Degree $2$ Conductor $8470$ Sign $-1$ Analytic cond. $67.6332$ Root an. cond. $8.22394$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 2-s + 4-s + 5-s + 7-s + 8-s − 3·9-s + 10-s − 2·13-s + 14-s + 16-s − 6·17-s − 3·18-s − 4·19-s + 20-s + 4·23-s + 25-s − 2·26-s + 28-s + 2·29-s + 8·31-s + 32-s − 6·34-s + 35-s − 3·36-s − 10·37-s − 4·38-s + 40-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s − 9-s + 0.316·10-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s − 0.917·19-s + 0.223·20-s + 0.834·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s + 0.371·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s + 0.169·35-s − 1/2·36-s − 1.64·37-s − 0.648·38-s + 0.158·40-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$8470$$    =    $$2 \cdot 5 \cdot 7 \cdot 11^{2}$$ Sign: $-1$ Analytic conductor: $$67.6332$$ Root analytic conductor: $$8.22394$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{8470} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 8470,\ (\ :1/2),\ -1)$$

## 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 - T$$
5 $$1 - T$$
7 $$1 - T$$
11 $$1$$
good3 $$1 + p T^{2}$$
13 $$1 + 2 T + p T^{2}$$
17 $$1 + 6 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 + 10 T + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 + 12 T + p T^{2}$$
47 $$1 - 12 T + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 + 12 T + p T^{2}$$
61 $$1 + 6 T + p T^{2}$$
67 $$1 - 8 T + p T^{2}$$
71 $$1 + 8 T + p T^{2}$$
73 $$1 + 14 T + p T^{2}$$
79 $$1 + p T^{2}$$
83 $$1 + 4 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}$$