# Properties

 Label 8-882e4-1.1-c1e4-0-0 Degree $8$ Conductor $605165749776$ Sign $1$ Analytic cond. $2460.26$ Root an. cond. $2.65382$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s + 4-s + 2·8-s − 8·11-s − 4·16-s + 16·22-s − 16·23-s + 8·25-s + 8·29-s + 2·32-s − 8·37-s − 16·43-s − 8·44-s + 32·46-s − 16·50-s − 8·53-s − 16·58-s + 3·64-s + 24·67-s + 16·74-s + 32·79-s + 32·86-s − 16·88-s − 16·92-s + 8·100-s + 16·106-s + 8·107-s + ⋯
 L(s)  = 1 − 1.41·2-s + 1/2·4-s + 0.707·8-s − 2.41·11-s − 16-s + 3.41·22-s − 3.33·23-s + 8/5·25-s + 1.48·29-s + 0.353·32-s − 1.31·37-s − 2.43·43-s − 1.20·44-s + 4.71·46-s − 2.26·50-s − 1.09·53-s − 2.10·58-s + 3/8·64-s + 2.93·67-s + 1.85·74-s + 3.60·79-s + 3.45·86-s − 1.70·88-s − 1.66·92-s + 4/5·100-s + 1.55·106-s + 0.773·107-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{8} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$2460.26$$ Root analytic conductor: $$2.65382$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{882} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.1029839502$$ $$L(\frac12)$$ $$\approx$$ $$0.1029839502$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ $$( 1 + T + T^{2} )^{2}$$
3 $$1$$
7 $$1$$
good5$C_2^3$ $$1 - 8 T^{2} + 39 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8}$$
11$C_2^2$ $$( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
13$C_2^2$ $$( 1 + 8 T^{2} + p^{2} T^{4} )^{2}$$
17$C_2^3$ $$1 + 16 T^{2} - 33 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8}$$
19$C_2^3$ $$1 - 6 T^{2} - 325 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8}$$
23$C_2^2$ $$( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
29$C_2$ $$( 1 - 2 T + p T^{2} )^{4}$$
31$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
37$C_2^2$ $$( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2^2$ $$( 1 - 16 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2$ $$( 1 + 4 T + p T^{2} )^{4}$$
47$C_2^3$ $$1 - 62 T^{2} + 1635 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2^2$ $$( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
59$C_2^2$$\times$$C_2^2$ $$( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 167 T^{2} + 18 p T^{3} + p^{2} T^{4} )$$
61$C_2^3$ $$1 - 120 T^{2} + 10679 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8}$$
67$C_2^2$ $$( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_2$ $$( 1 + p T^{2} )^{4}$$
73$C_2^3$ $$1 + 96 T^{2} + 3887 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8}$$
79$C_2^2$ $$( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}$$
83$C_2^2$ $$( 1 + 134 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^3$ $$1 - 128 T^{2} + 8463 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8}$$
97$C_2^2$ $$( 1 + 144 T^{2} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$