# Properties

 Label 8-21e8-1.1-c5e4-0-4 Degree $8$ Conductor $37822859361$ Sign $1$ Analytic cond. $2.50262\times 10^{7}$ Root an. cond. $8.41006$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 10·2-s − 9·4-s − 440·8-s + 1.95e3·11-s − 1.43e3·16-s + 1.95e4·22-s + 7.13e3·23-s − 4.86e3·25-s + 3.35e3·29-s − 3.45e3·32-s − 9.20e3·37-s + 2.04e4·43-s − 1.75e4·44-s + 7.13e4·46-s − 4.86e4·50-s + 1.02e5·53-s + 3.35e4·58-s − 3.02e4·64-s − 2.28e4·67-s + 1.53e5·71-s − 9.20e4·74-s − 9.06e4·79-s + 2.04e5·86-s − 8.58e5·88-s − 6.42e4·92-s + 4.38e4·100-s + 1.02e6·106-s + ⋯
 L(s)  = 1 + 1.76·2-s − 0.281·4-s − 2.43·8-s + 4.86·11-s − 1.39·16-s + 8.59·22-s + 2.81·23-s − 1.55·25-s + 0.740·29-s − 0.595·32-s − 1.10·37-s + 1.68·43-s − 1.36·44-s + 4.97·46-s − 2.75·50-s + 5.03·53-s + 1.30·58-s − 0.922·64-s − 0.623·67-s + 3.62·71-s − 1.95·74-s − 1.63·79-s + 2.98·86-s − 11.8·88-s − 0.791·92-s + 0.438·100-s + 8.89·106-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{8} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$2.50262\times 10^{7}$$ Root analytic conductor: $$8.41006$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{8} \cdot 7^{8} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$24.07283228$$ $$L(\frac12)$$ $$\approx$$ $$24.07283228$$ $$L(\frac{7}{2})$$ 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)$
bad3 $$1$$
7 $$1$$
good2$D_{4}$ $$( 1 - 5 T + 21 p T^{2} - 5 p^{5} T^{3} + p^{10} T^{4} )^{2}$$
5$D_4\times C_2$ $$1 + 4868 T^{2} + 22562806 T^{4} + 4868 p^{10} T^{6} + p^{20} T^{8}$$
11$D_{4}$ $$( 1 - 976 T + 556178 T^{2} - 976 p^{5} T^{3} + p^{10} T^{4} )^{2}$$
13$D_4\times C_2$ $$1 + 224500 T^{2} - 32848089674 T^{4} + 224500 p^{10} T^{6} + p^{20} T^{8}$$
17$D_4\times C_2$ $$1 + 3065760 T^{2} + 6374095942466 T^{4} + 3065760 p^{10} T^{6} + p^{20} T^{8}$$
19$D_4\times C_2$ $$1 + 8544504 T^{2} + 30052000046306 T^{4} + 8544504 p^{10} T^{6} + p^{20} T^{8}$$
23$D_{4}$ $$( 1 - 3568 T + 12712350 T^{2} - 3568 p^{5} T^{3} + p^{10} T^{4} )^{2}$$
29$D_{4}$ $$( 1 - 1676 T + 24360510 T^{2} - 1676 p^{5} T^{3} + p^{10} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 + 29395756 T^{2} + 658228795954534 T^{4} + 29395756 p^{10} T^{6} + p^{20} T^{8}$$
37$D_{4}$ $$( 1 + 4604 T + 143922030 T^{2} + 4604 p^{5} T^{3} + p^{10} T^{4} )^{2}$$
41$D_4\times C_2$ $$1 + 212331360 T^{2} + 36724946155085474 T^{4} + 212331360 p^{10} T^{6} + p^{20} T^{8}$$
43$D_{4}$ $$( 1 - 10224 T + 293018130 T^{2} - 10224 p^{5} T^{3} + p^{10} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 + 568199404 T^{2} + 172542701189040294 T^{4} + 568199404 p^{10} T^{6} + p^{20} T^{8}$$
53$D_{4}$ $$( 1 - 51460 T + 1308627278 T^{2} - 51460 p^{5} T^{3} + p^{10} T^{4} )^{2}$$
59$D_4\times C_2$ $$1 + 1609944152 T^{2} + 1391734325057519170 T^{4} + 1609944152 p^{10} T^{6} + p^{20} T^{8}$$
61$D_4\times C_2$ $$1 + 1094138468 T^{2} + 1609346124964468758 T^{4} + 1094138468 p^{10} T^{6} + p^{20} T^{8}$$
67$D_{4}$ $$( 1 + 11448 T - 1089307338 T^{2} + 11448 p^{5} T^{3} + p^{10} T^{4} )^{2}$$
71$D_{4}$ $$( 1 - 76912 T + 4562060670 T^{2} - 76912 p^{5} T^{3} + p^{10} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 + 2170565248 T^{2} + 425171557315203874 T^{4} + 2170565248 p^{10} T^{6} + p^{20} T^{8}$$
79$D_{4}$ $$( 1 + 45344 T + 6154499470 T^{2} + 45344 p^{5} T^{3} + p^{10} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 + 6721792472 T^{2} + 21939991980485194594 T^{4} + 6721792472 p^{10} T^{6} + p^{20} T^{8}$$
89$D_4\times C_2$ $$1 + 14718300768 T^{2} +$$$$11\!\cdots\!58$$$$T^{4} + 14718300768 p^{10} T^{6} + p^{20} T^{8}$$
97$D_4\times C_2$ $$1 + 22489794400 T^{2} + 25388976518139394 p^{2} T^{4} + 22489794400 p^{10} T^{6} + p^{20} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$