# Properties

 Label 8-210e4-1.1-c3e4-0-3 Degree $8$ Conductor $1944810000$ Sign $1$ Analytic cond. $23569.0$ Root an. cond. $3.52000$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·2-s + 6·3-s + 4·4-s + 10·5-s − 24·6-s + 24·7-s + 16·8-s + 9·9-s − 40·10-s − 50·11-s + 24·12-s − 200·13-s − 96·14-s + 60·15-s − 64·16-s − 34·17-s − 36·18-s + 46·19-s + 40·20-s + 144·21-s + 200·22-s − 126·23-s + 96·24-s + 25·25-s + 800·26-s − 54·27-s + 96·28-s + ⋯
 L(s)  = 1 − 1.41·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s − 1.63·6-s + 1.29·7-s + 0.707·8-s + 1/3·9-s − 1.26·10-s − 1.37·11-s + 0.577·12-s − 4.26·13-s − 1.83·14-s + 1.03·15-s − 16-s − 0.485·17-s − 0.471·18-s + 0.555·19-s + 0.447·20-s + 1.49·21-s + 1.93·22-s − 1.14·23-s + 0.816·24-s + 1/5·25-s + 6.03·26-s − 0.384·27-s + 0.647·28-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$23569.0$$ Root analytic conductor: $$3.52000$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{210} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.364894247$$ $$L(\frac12)$$ $$\approx$$ $$1.364894247$$ $$L(\frac{5}{2})$$ 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$C_2$ $$( 1 + p T + p^{2} T^{2} )^{2}$$
3$C_2$ $$( 1 - p T + p^{2} T^{2} )^{2}$$
5$C_2$ $$( 1 - p T + p^{2} T^{2} )^{2}$$
7$C_2^2$ $$1 - 24 T + 535 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4}$$
good11$D_4\times C_2$ $$1 + 50 T - 492 T^{2} + 1500 p T^{3} + 31843 p^{2} T^{4} + 1500 p^{4} T^{5} - 492 p^{6} T^{6} + 50 p^{9} T^{7} + p^{12} T^{8}$$
13$D_{4}$ $$( 1 + 100 T + 6599 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 + 2 p T - 1584 T^{2} - 14172 p T^{3} - 22309397 T^{4} - 14172 p^{4} T^{5} - 1584 p^{6} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8}$$
19$D_4\times C_2$ $$1 - 46 T - 7411 T^{2} + 192786 T^{3} + 29204204 T^{4} + 192786 p^{3} T^{5} - 7411 p^{6} T^{6} - 46 p^{9} T^{7} + p^{12} T^{8}$$
23$D_4\times C_2$ $$1 + 126 T + 11468 T^{2} - 2510676 T^{3} - 321768717 T^{4} - 2510676 p^{3} T^{5} + 11468 p^{6} T^{6} + 126 p^{9} T^{7} + p^{12} T^{8}$$
29$D_{4}$ $$( 1 + 150 T + 51748 T^{2} + 150 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 - 74 T - 12995 T^{2} + 3042214 T^{3} - 709461356 T^{4} + 3042214 p^{3} T^{5} - 12995 p^{6} T^{6} - 74 p^{9} T^{7} + p^{12} T^{8}$$
37$D_4\times C_2$ $$1 - 176 T - 54179 T^{2} + 2842576 T^{3} + 3116620288 T^{4} + 2842576 p^{3} T^{5} - 54179 p^{6} T^{6} - 176 p^{9} T^{7} + p^{12} T^{8}$$
41$D_{4}$ $$( 1 - 10 T + 52612 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
43$D_{4}$ $$( 1 - 132 T + 148915 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 + 316 T - 128034 T^{2} + 6397104 T^{3} + 30787287283 T^{4} + 6397104 p^{3} T^{5} - 128034 p^{6} T^{6} + 316 p^{9} T^{7} + p^{12} T^{8}$$
53$D_4\times C_2$ $$1 + 236 T - 226482 T^{2} - 3675936 T^{3} + 49168209163 T^{4} - 3675936 p^{3} T^{5} - 226482 p^{6} T^{6} + 236 p^{9} T^{7} + p^{12} T^{8}$$
59$D_4\times C_2$ $$1 - 58 T + 300060 T^{2} + 41032332 T^{3} + 45157087819 T^{4} + 41032332 p^{3} T^{5} + 300060 p^{6} T^{6} - 58 p^{9} T^{7} + p^{12} T^{8}$$
61$D_4\times C_2$ $$1 - 792 T + 21206 T^{2} - 120460032 T^{3} + 173408915019 T^{4} - 120460032 p^{3} T^{5} + 21206 p^{6} T^{6} - 792 p^{9} T^{7} + p^{12} T^{8}$$
67$D_4\times C_2$ $$1 - 868 T + 13397 T^{2} - 120218868 T^{3} + 230579516048 T^{4} - 120218868 p^{3} T^{5} + 13397 p^{6} T^{6} - 868 p^{9} T^{7} + p^{12} T^{8}$$
71$D_{4}$ $$( 1 + 386 T + 568696 T^{2} + 386 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 - 1220 T + 404641 T^{2} - 372984500 T^{3} + 444186440992 T^{4} - 372984500 p^{3} T^{5} + 404641 p^{6} T^{6} - 1220 p^{9} T^{7} + p^{12} T^{8}$$
79$D_4\times C_2$ $$1 + 54 T - 412771 T^{2} - 30801114 T^{3} - 71729475276 T^{4} - 30801114 p^{3} T^{5} - 412771 p^{6} T^{6} + 54 p^{9} T^{7} + p^{12} T^{8}$$
83$D_{4}$ $$( 1 + 182 T + 54160 T^{2} + 182 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 - 418 T - 1122840 T^{2} + 46972332 T^{3} + 1063516039579 T^{4} + 46972332 p^{3} T^{5} - 1122840 p^{6} T^{6} - 418 p^{9} T^{7} + p^{12} T^{8}$$
97$D_{4}$ $$( 1 + 392 T + 1821282 T^{2} + 392 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$