# Properties

 Label 2-8568-1.1-c1-0-108 Degree $2$ Conductor $8568$ Sign $-1$ Analytic cond. $68.4158$ Root an. cond. $8.27138$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.40·5-s − 7-s + 1.23·11-s − 6.47·13-s − 17-s − 2.21·19-s − 1.39·23-s + 6.60·25-s + 0.633·29-s + 0.965·31-s − 3.40·35-s − 8.31·37-s + 5.60·41-s + 11.0·43-s − 7.67·47-s + 49-s − 11.7·53-s + 4.21·55-s + 0.602·59-s − 9.84·61-s − 22.0·65-s + 5.30·67-s + 3.33·71-s + 14.0·73-s − 1.23·77-s − 0.323·79-s − 13.8·83-s + ⋯
 L(s)  = 1 + 1.52·5-s − 0.377·7-s + 0.372·11-s − 1.79·13-s − 0.242·17-s − 0.507·19-s − 0.291·23-s + 1.32·25-s + 0.117·29-s + 0.173·31-s − 0.575·35-s − 1.36·37-s + 0.875·41-s + 1.68·43-s − 1.11·47-s + 0.142·49-s − 1.60·53-s + 0.567·55-s + 0.0783·59-s − 1.26·61-s − 2.73·65-s + 0.648·67-s + 0.395·71-s + 1.64·73-s − 0.140·77-s − 0.0363·79-s − 1.52·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$8568$$    =    $$2^{3} \cdot 3^{2} \cdot 7 \cdot 17$$ Sign: $-1$ Analytic conductor: $$68.4158$$ Root analytic conductor: $$8.27138$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 8568,\ (\ :1/2),\ -1)$$

## 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$$
3 $$1$$
7 $$1 + T$$
17 $$1 + T$$
good5 $$1 - 3.40T + 5T^{2}$$
11 $$1 - 1.23T + 11T^{2}$$
13 $$1 + 6.47T + 13T^{2}$$
19 $$1 + 2.21T + 19T^{2}$$
23 $$1 + 1.39T + 23T^{2}$$
29 $$1 - 0.633T + 29T^{2}$$
31 $$1 - 0.965T + 31T^{2}$$
37 $$1 + 8.31T + 37T^{2}$$
41 $$1 - 5.60T + 41T^{2}$$
43 $$1 - 11.0T + 43T^{2}$$
47 $$1 + 7.67T + 47T^{2}$$
53 $$1 + 11.7T + 53T^{2}$$
59 $$1 - 0.602T + 59T^{2}$$
61 $$1 + 9.84T + 61T^{2}$$
67 $$1 - 5.30T + 67T^{2}$$
71 $$1 - 3.33T + 71T^{2}$$
73 $$1 - 14.0T + 73T^{2}$$
79 $$1 + 0.323T + 79T^{2}$$
83 $$1 + 13.8T + 83T^{2}$$
89 $$1 - 14.7T + 89T^{2}$$
97 $$1 + 5.90T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$