# Properties

 Label 8-882e4-1.1-c2e4-0-3 Degree $8$ Conductor $605165749776$ Sign $1$ Analytic cond. $333591.$ Root an. cond. $4.90232$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·4-s − 24·11-s + 12·16-s + 48·23-s + 28·25-s − 44·37-s + 28·43-s − 96·44-s + 240·53-s + 32·64-s − 220·67-s − 312·71-s + 20·79-s + 192·92-s + 112·100-s − 576·107-s + 580·109-s − 168·113-s − 124·121-s + 127-s + 131-s + 137-s + 139-s − 176·148-s + 149-s + 151-s + 157-s + ⋯
 L(s)  = 1 + 4-s − 2.18·11-s + 3/4·16-s + 2.08·23-s + 1.11·25-s − 1.18·37-s + 0.651·43-s − 2.18·44-s + 4.52·53-s + 1/2·64-s − 3.28·67-s − 4.39·71-s + 0.253·79-s + 2.08·92-s + 1.11·100-s − 5.38·107-s + 5.32·109-s − 1.48·113-s − 1.02·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1.18·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-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(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{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: $$333591.$$ Root analytic conductor: $$4.90232$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{882} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.1923482081$$ $$L(\frac12)$$ $$\approx$$ $$0.1923482081$$ $$L(2)$$ 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$C_2$ $$( 1 - p T^{2} )^{2}$$
3 $$1$$
7 $$1$$
good5$D_4\times C_2$ $$1 - 28 T^{2} + 294 T^{4} - 28 p^{4} T^{6} + p^{8} T^{8}$$
11$C_2$ $$( 1 + 6 T + p^{2} T^{2} )^{4}$$
13$D_4\times C_2$ $$1 + 98 T^{2} + 54915 T^{4} + 98 p^{4} T^{6} + p^{8} T^{8}$$
17$D_4\times C_2$ $$1 - 724 T^{2} + 279654 T^{4} - 724 p^{4} T^{6} + p^{8} T^{8}$$
19$D_4\times C_2$ $$1 - 58 p T^{2} + 550131 T^{4} - 58 p^{5} T^{6} + p^{8} T^{8}$$
23$D_{4}$ $$( 1 - 24 T + 554 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
29$C_2^2$ $$( 1 + 530 T^{2} + p^{4} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 - 1678 T^{2} + 1875 p^{2} T^{4} - 1678 p^{4} T^{6} + p^{8} T^{8}$$
37$D_{4}$ $$( 1 + 22 T + 2571 T^{2} + 22 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
41$D_4\times C_2$ $$1 - 2476 T^{2} + 6405414 T^{4} - 2476 p^{4} T^{6} + p^{8} T^{8}$$
43$D_{4}$ $$( 1 - 14 T + 3675 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 - 5884 T^{2} + 18275334 T^{4} - 5884 p^{4} T^{6} + p^{8} T^{8}$$
53$D_{4}$ $$( 1 - 120 T + 8570 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
59$D_4\times C_2$ $$1 - 11476 T^{2} + 56933574 T^{4} - 11476 p^{4} T^{6} + p^{8} T^{8}$$
61$D_4\times C_2$ $$1 - 13252 T^{2} + 70932006 T^{4} - 13252 p^{4} T^{6} + p^{8} T^{8}$$
67$D_{4}$ $$( 1 + 110 T + 8475 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
71$D_{4}$ $$( 1 + 156 T + 178 p T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 - 1390 T^{2} + 43340307 T^{4} - 1390 p^{4} T^{6} + p^{8} T^{8}$$
79$D_{4}$ $$( 1 - 10 T + 3795 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 116 p T^{2} + 107694438 T^{4} - 116 p^{5} T^{6} + p^{8} T^{8}$$
89$C_2^2$ $$( 1 - 15410 T^{2} + p^{4} T^{4} )^{2}$$
97$D_4\times C_2$ $$1 - 36580 T^{2} + 511416774 T^{4} - 36580 p^{4} T^{6} + p^{8} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$