# Properties

 Label 8-2842e4-1.1-c1e4-0-2 Degree $8$ Conductor $6.524\times 10^{13}$ Sign $1$ Analytic cond. $265219.$ Root an. cond. $4.76376$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·2-s + 10·4-s − 20·8-s − 5·9-s − 10·11-s + 35·16-s + 20·18-s + 40·22-s − 15·25-s + 4·29-s − 56·32-s − 50·36-s + 12·37-s + 6·43-s − 100·44-s + 60·50-s − 38·53-s − 16·58-s + 84·64-s − 36·67-s − 4·71-s + 100·72-s − 48·74-s − 10·79-s + 5·81-s − 24·86-s + 200·88-s + ⋯
 L(s)  = 1 − 2.82·2-s + 5·4-s − 7.07·8-s − 5/3·9-s − 3.01·11-s + 35/4·16-s + 4.71·18-s + 8.52·22-s − 3·25-s + 0.742·29-s − 9.89·32-s − 8.33·36-s + 1.97·37-s + 0.914·43-s − 15.0·44-s + 8.48·50-s − 5.21·53-s − 2.10·58-s + 21/2·64-s − 4.39·67-s − 0.474·71-s + 11.7·72-s − 5.57·74-s − 1.12·79-s + 5/9·81-s − 2.58·86-s + 21.3·88-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 29^{4}\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 7^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 7^{8} \cdot 29^{4}$$ Sign: $1$ Analytic conductor: $$265219.$$ Root analytic conductor: $$4.76376$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 2^{4} \cdot 7^{8} \cdot 29^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## 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$C_1$ $$( 1 + T )^{4}$$
7 $$1$$
29$C_1$ $$( 1 - T )^{4}$$
good3$C_2^2:C_4$ $$1 + 5 T^{2} + 20 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8}$$
5$C_2^2 \wr C_2$ $$1 + 3 p T^{2} + 102 T^{4} + 3 p^{3} T^{6} + p^{4} T^{8}$$
11$D_{4}$ $$( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2}$$
13$C_2^2 \wr C_2$ $$1 + 15 T^{2} + 30 p T^{4} + 15 p^{2} T^{6} + p^{4} T^{8}$$
17$C_2^2 \wr C_2$ $$1 + 22 T^{2} + 682 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8}$$
19$C_2^2 \wr C_2$ $$1 - 16 T^{2} + 718 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8}$$
23$C_2^2$ $$( 1 - 22 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2^2 \wr C_2$ $$1 + 87 T^{2} + 3810 T^{4} + 87 p^{2} T^{6} + p^{4} T^{8}$$
37$D_{4}$ $$( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2^2 \wr C_2$ $$1 + 118 T^{2} + 6826 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8}$$
43$D_{4}$ $$( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
47$C_2^2 \wr C_2$ $$1 + 183 T^{2} + 12786 T^{4} + 183 p^{2} T^{6} + p^{4} T^{8}$$
53$D_{4}$ $$( 1 + 19 T + 192 T^{2} + 19 p T^{3} + p^{2} T^{4} )^{2}$$
59$C_2^2 \wr C_2$ $$1 + 190 T^{2} + 15970 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8}$$
61$C_2^2 \wr C_2$ $$1 + 224 T^{2} + 19918 T^{4} + 224 p^{2} T^{6} + p^{4} T^{8}$$
67$D_{4}$ $$( 1 + 18 T + 198 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2}$$
71$D_{4}$ $$( 1 + 2 T - 10 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
73$C_2^2 \wr C_2$ $$1 - 58 T^{2} + 874 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8}$$
79$D_{4}$ $$( 1 + 5 T + 58 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2}$$
83$C_2^2 \wr C_2$ $$1 + 206 T^{2} + 23010 T^{4} + 206 p^{2} T^{6} + p^{4} T^{8}$$
89$C_2^2 \wr C_2$ $$1 + 266 T^{2} + 32154 T^{4} + 266 p^{2} T^{6} + p^{4} T^{8}$$
97$C_2^2 \wr C_2$ $$1 + 150 T^{2} + 19530 T^{4} + 150 p^{2} T^{6} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$