# Properties

 Label 8-4840e4-1.1-c1e4-0-1 Degree $8$ Conductor $5.488\times 10^{14}$ Sign $1$ Analytic cond. $2.23095\times 10^{6}$ Root an. cond. $6.21671$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·3-s − 4·5-s − 6·7-s − 2·9-s − 4·13-s + 8·15-s + 12·21-s + 4·23-s + 10·25-s + 8·27-s − 16·29-s + 16·31-s + 24·35-s − 8·37-s + 8·39-s − 8·41-s + 10·43-s + 8·45-s + 14·47-s + 2·49-s + 8·53-s + 4·61-s + 12·63-s + 16·65-s − 2·67-s − 8·69-s + 8·71-s + ⋯
 L(s)  = 1 − 1.15·3-s − 1.78·5-s − 2.26·7-s − 2/3·9-s − 1.10·13-s + 2.06·15-s + 2.61·21-s + 0.834·23-s + 2·25-s + 1.53·27-s − 2.97·29-s + 2.87·31-s + 4.05·35-s − 1.31·37-s + 1.28·39-s − 1.24·41-s + 1.52·43-s + 1.19·45-s + 2.04·47-s + 2/7·49-s + 1.09·53-s + 0.512·61-s + 1.51·63-s + 1.98·65-s − 0.244·67-s − 0.963·69-s + 0.949·71-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{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^{12} \cdot 5^{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: $$2^{12} \cdot 5^{4} \cdot 11^{8}$$ Sign: $1$ Analytic conductor: $$2.23095\times 10^{6}$$ Root analytic conductor: $$6.21671$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 2^{12} \cdot 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 $$1$$
5$C_1$ $$( 1 + T )^{4}$$
11 $$1$$
good3$C_2 \wr C_2\wr C_2$ $$1 + 2 T + 2 p T^{2} + 8 T^{3} + 19 T^{4} + 8 p T^{5} + 2 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
7$C_2 \wr C_2\wr C_2$ $$1 + 6 T + 34 T^{2} + 120 T^{3} + 375 T^{4} + 120 p T^{5} + 34 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2 \wr C_2\wr C_2$ $$1 + 4 T + 40 T^{2} + 124 T^{3} + 718 T^{4} + 124 p T^{5} + 40 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2$ $$( 1 + p T^{2} )^{4}$$
19$C_2 \wr C_2\wr C_2$ $$1 + 40 T^{2} - 96 T^{3} + 750 T^{4} - 96 p T^{5} + 40 p^{2} T^{6} + p^{4} T^{8}$$
23$C_2 \wr C_2\wr C_2$ $$1 - 4 T + 56 T^{2} - 292 T^{3} + 1534 T^{4} - 292 p T^{5} + 56 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2$ $$( 1 + 4 T + p T^{2} )^{4}$$
31$C_2 \wr C_2\wr C_2$ $$1 - 16 T + 160 T^{2} - 1120 T^{3} + 6862 T^{4} - 1120 p T^{5} + 160 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2 \wr C_2\wr C_2$ $$1 + 8 T + 100 T^{2} + 632 T^{3} + 4918 T^{4} + 632 p T^{5} + 100 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr C_2\wr C_2$ $$1 + 8 T + 134 T^{2} + 992 T^{3} + 7627 T^{4} + 992 p T^{5} + 134 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr C_2\wr C_2$ $$1 - 10 T + 118 T^{2} - 784 T^{3} + 6571 T^{4} - 784 p T^{5} + 118 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr C_2\wr C_2$ $$1 - 14 T + 194 T^{2} - 1640 T^{3} + 13975 T^{4} - 1640 p T^{5} + 194 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr C_2\wr C_2$ $$1 - 8 T + 200 T^{2} - 1256 T^{3} + 15598 T^{4} - 1256 p T^{5} + 200 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr C_2\wr C_2$ $$1 + 128 T^{2} - 432 T^{3} + 7710 T^{4} - 432 p T^{5} + 128 p^{2} T^{6} + p^{4} T^{8}$$
61$C_2 \wr C_2\wr C_2$ $$1 - 4 T + 154 T^{2} - 832 T^{3} + 11575 T^{4} - 832 p T^{5} + 154 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr C_2\wr C_2$ $$1 + 2 T + 166 T^{2} + 632 T^{3} + 13843 T^{4} + 632 p T^{5} + 166 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr C_2\wr C_2$ $$1 - 8 T + 128 T^{2} - 1496 T^{3} + 11758 T^{4} - 1496 p T^{5} + 128 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2$ $$( 1 + 8 T + p T^{2} )^{4}$$
79$C_2 \wr C_2\wr C_2$ $$1 - 4 T + 196 T^{2} - 292 T^{3} + 17734 T^{4} - 292 p T^{5} + 196 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr C_2\wr C_2$ $$1 + 12 T + 152 T^{2} + 924 T^{3} + 6798 T^{4} + 924 p T^{5} + 152 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr C_2\wr C_2$ $$1 - 12 T + 266 T^{2} - 1680 T^{3} + 27219 T^{4} - 1680 p T^{5} + 266 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr C_2\wr C_2$ $$1 + 184 T^{2} + 528 T^{3} + 20286 T^{4} + 528 p T^{5} + 184 p^{2} T^{6} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$