# Properties

 Label 8-210e4-1.1-c3e4-0-1 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 − 8·4-s − 4·5-s − 18·9-s + 24·11-s + 48·16-s + 8·19-s + 32·20-s + 146·25-s − 40·29-s − 584·31-s + 144·36-s − 264·41-s − 192·44-s + 72·45-s − 98·49-s − 96·55-s − 448·59-s − 24·61-s − 256·64-s − 312·71-s − 64·76-s − 752·79-s − 192·80-s + 243·81-s − 1.09e3·89-s − 32·95-s − 432·99-s + ⋯
 L(s)  = 1 − 4-s − 0.357·5-s − 2/3·9-s + 0.657·11-s + 3/4·16-s + 0.0965·19-s + 0.357·20-s + 1.16·25-s − 0.256·29-s − 3.38·31-s + 2/3·36-s − 1.00·41-s − 0.657·44-s + 0.238·45-s − 2/7·49-s − 0.235·55-s − 0.988·59-s − 0.0503·61-s − 1/2·64-s − 0.521·71-s − 0.0965·76-s − 1.07·79-s − 0.268·80-s + 1/3·81-s − 1.30·89-s − 0.0345·95-s − 0.438·99-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

 $$L(2)$$ $$\approx$$ $$0.07769183680$$ $$L(\frac12)$$ $$\approx$$ $$0.07769183680$$ $$L(\frac{5}{2})$$ 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$C_2$ $$( 1 + p^{2} T^{2} )^{2}$$
3$C_2$ $$( 1 + p^{2} T^{2} )^{2}$$
5$C_2^2$ $$1 + 4 T - 26 p T^{2} + 4 p^{3} T^{3} + p^{6} T^{4}$$
7$C_2$ $$( 1 + p^{2} T^{2} )^{2}$$
good11$D_{4}$ $$( 1 - 12 T + 2314 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
13$D_4\times C_2$ $$1 - 716 T^{2} - 5578218 T^{4} - 716 p^{6} T^{6} + p^{12} T^{8}$$
17$D_4\times C_2$ $$1 - 9540 T^{2} + 48909638 T^{4} - 9540 p^{6} T^{6} + p^{12} T^{8}$$
19$D_{4}$ $$( 1 - 4 T + 13626 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 - 14172 T^{2} + 279829670 T^{4} - 14172 p^{6} T^{6} + p^{12} T^{8}$$
29$D_{4}$ $$( 1 + 20 T + 48782 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
31$D_{4}$ $$( 1 + 292 T + 80514 T^{2} + 292 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
37$D_4\times C_2$ $$1 - 122420 T^{2} + 7280062518 T^{4} - 122420 p^{6} T^{6} + p^{12} T^{8}$$
41$D_{4}$ $$( 1 + 132 T + 136054 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 - 82316 T^{2} + 14216284662 T^{4} - 82316 p^{6} T^{6} + p^{12} T^{8}$$
47$D_4\times C_2$ $$1 - 198492 T^{2} + 22457318918 T^{4} - 198492 p^{6} T^{6} + p^{12} T^{8}$$
53$D_4\times C_2$ $$1 - 110828 T^{2} + 21717634998 T^{4} - 110828 p^{6} T^{6} + p^{12} T^{8}$$
59$D_{4}$ $$( 1 + 224 T + 30086 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
61$D_{4}$ $$( 1 + 12 T + 292622 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
67$D_4\times C_2$ $$1 - 487820 T^{2} + 116820910038 T^{4} - 487820 p^{6} T^{6} + p^{12} T^{8}$$
71$D_{4}$ $$( 1 + 156 T + 491410 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 - 1463324 T^{2} + 835923760038 T^{4} - 1463324 p^{6} T^{6} + p^{12} T^{8}$$
79$D_{4}$ $$( 1 + 376 T + 1015278 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 125228 T^{2} + 394047302358 T^{4} - 125228 p^{6} T^{6} + p^{12} T^{8}$$
89$D_{4}$ $$( 1 + 548 T + 237398 T^{2} + 548 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
97$D_4\times C_2$ $$1 - 2031292 T^{2} + 2294372967174 T^{4} - 2031292 p^{6} T^{6} + p^{12} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$