# Properties

 Label 8-8470e4-1.1-c1e4-0-1 Degree $8$ Conductor $5.147\times 10^{15}$ Sign $1$ Analytic cond. $2.09238\times 10^{7}$ Root an. cond. $8.22394$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·2-s + 2·3-s + 10·4-s − 4·5-s + 8·6-s + 4·7-s + 20·8-s − 9-s − 16·10-s + 20·12-s + 13-s + 16·14-s − 8·15-s + 35·16-s + 2·17-s − 4·18-s − 3·19-s − 40·20-s + 8·21-s − 8·23-s + 40·24-s + 10·25-s + 4·26-s − 2·27-s + 40·28-s + 15·29-s − 32·30-s + ⋯
 L(s)  = 1 + 2.82·2-s + 1.15·3-s + 5·4-s − 1.78·5-s + 3.26·6-s + 1.51·7-s + 7.07·8-s − 1/3·9-s − 5.05·10-s + 5.77·12-s + 0.277·13-s + 4.27·14-s − 2.06·15-s + 35/4·16-s + 0.485·17-s − 0.942·18-s − 0.688·19-s − 8.94·20-s + 1.74·21-s − 1.66·23-s + 8.16·24-s + 2·25-s + 0.784·26-s − 0.384·27-s + 7.55·28-s + 2.78·29-s − 5.84·30-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{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 5^{4} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8}$$ Sign: $1$ Analytic conductor: $$2.09238\times 10^{7}$$ Root analytic conductor: $$8.22394$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{8470} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$122.5242991$$ $$L(\frac12)$$ $$\approx$$ $$122.5242991$$ $$L(\frac{3}{2})$$ 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$C_1$ $$( 1 - T )^{4}$$
5$C_1$ $$( 1 + T )^{4}$$
7$C_1$ $$( 1 - T )^{4}$$
11 $$1$$
good3$C_2 \wr C_2\wr C_2$ $$1 - 2 T + 5 T^{2} - 10 T^{3} + 23 T^{4} - 10 p T^{5} + 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2 \wr C_2\wr C_2$ $$1 - T + 3 p T^{2} - 23 T^{3} + 672 T^{4} - 23 p T^{5} + 3 p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
17$C_2 \wr C_2\wr C_2$ $$1 - 2 T + 3 p T^{2} - 64 T^{3} + 1137 T^{4} - 64 p T^{5} + 3 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2 \wr C_2\wr C_2$ $$1 + 3 T + 35 T^{2} + 153 T^{3} + 844 T^{4} + 153 p T^{5} + 35 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2$ $$( 1 + 2 T + p T^{2} )^{4}$$
29$C_2 \wr C_2\wr C_2$ $$1 - 15 T + 177 T^{2} - 1395 T^{3} + 8628 T^{4} - 1395 p T^{5} + 177 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8}$$
31$D_{4}$ $$( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
37$C_2 \wr C_2\wr C_2$ $$1 - 2 T + 88 T^{2} - 46 T^{3} + 3838 T^{4} - 46 p T^{5} + 88 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr C_2\wr C_2$ $$1 - 11 T + 129 T^{2} - 1001 T^{3} + 8240 T^{4} - 1001 p T^{5} + 129 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr C_2\wr C_2$ $$1 - 7 T + 105 T^{2} - 605 T^{3} + 6408 T^{4} - 605 p T^{5} + 105 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr C_2\wr C_2$ $$1 - 5 T - 13 T^{2} - 195 T^{3} + 4964 T^{4} - 195 p T^{5} - 13 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr C_2\wr C_2$ $$1 - 10 T + 168 T^{2} - 30 p T^{3} + 12254 T^{4} - 30 p^{2} T^{5} + 168 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr C_2\wr C_2$ $$1 - 13 T + 151 T^{2} - 1427 T^{3} + 13660 T^{4} - 1427 p T^{5} + 151 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr C_2\wr C_2$ $$1 - 26 T + 444 T^{2} - 5086 T^{3} + 750 p T^{4} - 5086 p T^{5} + 444 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr C_2\wr C_2$ $$1 + 13 T + 287 T^{2} + 2415 T^{3} + 29196 T^{4} + 2415 p T^{5} + 287 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr C_2\wr C_2$ $$1 + 13 T + 233 T^{2} + 1991 T^{3} + 21640 T^{4} + 1991 p T^{5} + 233 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr C_2\wr C_2$ $$1 + 18 T + 267 T^{2} + 3288 T^{3} + 29193 T^{4} + 3288 p T^{5} + 267 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr C_2\wr C_2$ $$1 - 15 T + 387 T^{2} - 3675 T^{3} + 48708 T^{4} - 3675 p T^{5} + 387 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr C_2\wr C_2$ $$1 + 2 T + 217 T^{2} + 742 T^{3} + 22323 T^{4} + 742 p T^{5} + 217 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr C_2\wr C_2$ $$1 + 7 T + 251 T^{2} + 1223 T^{3} + 29920 T^{4} + 1223 p T^{5} + 251 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8}$$
97$D_{4}$ $$( 1 - 5 T + 49 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$