# Properties

 Label 8-4205e4-1.1-c1e4-0-0 Degree $8$ Conductor $3.127\times 10^{14}$ Sign $1$ Analytic cond. $1.27107\times 10^{6}$ Root an. cond. $5.79457$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·4-s + 4·5-s − 4·7-s − 8·9-s − 8·13-s + 7·16-s − 16·20-s − 12·23-s + 10·25-s + 16·28-s − 16·35-s + 32·36-s − 32·45-s − 12·49-s + 32·52-s + 24·59-s + 32·63-s − 8·64-s − 32·65-s + 4·67-s − 24·71-s + 28·80-s + 30·81-s − 12·83-s + 32·91-s + 48·92-s − 40·100-s + ⋯
 L(s)  = 1 − 2·4-s + 1.78·5-s − 1.51·7-s − 8/3·9-s − 2.21·13-s + 7/4·16-s − 3.57·20-s − 2.50·23-s + 2·25-s + 3.02·28-s − 2.70·35-s + 16/3·36-s − 4.77·45-s − 1.71·49-s + 4.43·52-s + 3.12·59-s + 4.03·63-s − 64-s − 3.96·65-s + 0.488·67-s − 2.84·71-s + 3.13·80-s + 10/3·81-s − 1.31·83-s + 3.35·91-s + 5.00·92-s − 4·100-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$5^{4} \cdot 29^{8}$$ Sign: $1$ Analytic conductor: $$1.27107\times 10^{6}$$ Root analytic conductor: $$5.79457$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 5^{4} \cdot 29^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## 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)$
bad5$C_1$ $$( 1 - T )^{4}$$
29 $$1$$
good2$D_4\times C_2$ $$1 + p^{2} T^{2} + 9 T^{4} + p^{4} T^{6} + p^{4} T^{8}$$
3$C_2^2$ $$( 1 + 4 T^{2} + p^{2} T^{4} )^{2}$$
7$D_{4}$ $$( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
11$D_4\times C_2$ $$1 + 16 T^{2} + 114 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8}$$
13$D_{4}$ $$( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 + 16 T^{2} + 450 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8}$$
19$C_2^2$ $$( 1 + 20 T^{2} + p^{2} T^{4} )^{2}$$
23$D_{4}$ $$( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
31$C_2^2$ $$( 1 + 44 T^{2} + p^{2} T^{4} )^{2}$$
37$C_2^2$ $$( 1 + 56 T^{2} + p^{2} T^{4} )^{2}$$
41$D_4\times C_2$ $$1 + 52 T^{2} + 3606 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8}$$
43$C_2^2$ $$( 1 + 68 T^{2} + p^{2} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 - 8 T^{2} + 78 p T^{4} - 8 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2$ $$( 1 + p T^{2} )^{4}$$
59$C_2$ $$( 1 - 6 T + p T^{2} )^{4}$$
61$D_4\times C_2$ $$1 + 100 T^{2} + 6054 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8}$$
67$D_{4}$ $$( 1 - 2 T - 12 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_2$ $$( 1 + 6 T + p T^{2} )^{4}$$
73$D_4\times C_2$ $$1 + 40 T^{2} - 4494 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8}$$
79$D_4\times C_2$ $$1 + 64 T^{2} - 2046 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8}$$
83$D_{4}$ $$( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 + 292 T^{2} + 36390 T^{4} + 292 p^{2} T^{6} + p^{4} T^{8}$$
97$C_2^2$ $$( 1 + 140 T^{2} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$