# Properties

 Label 8-280e4-1.1-c1e4-0-10 Degree $8$ Conductor $6146560000$ Sign $1$ Analytic cond. $24.9885$ Root an. cond. $1.49526$ 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 − 2·3-s + 8·4-s − 4·5-s + 8·6-s − 10·7-s − 8·8-s − 9-s + 16·10-s + 6·11-s − 16·12-s − 12·13-s + 40·14-s + 8·15-s − 4·16-s − 6·17-s + 4·18-s − 12·19-s − 32·20-s + 20·21-s − 24·22-s − 6·23-s + 16·24-s + 5·25-s + 48·26-s + 2·27-s − 80·28-s + ⋯
 L(s)  = 1 − 2.82·2-s − 1.15·3-s + 4·4-s − 1.78·5-s + 3.26·6-s − 3.77·7-s − 2.82·8-s − 1/3·9-s + 5.05·10-s + 1.80·11-s − 4.61·12-s − 3.32·13-s + 10.6·14-s + 2.06·15-s − 16-s − 1.45·17-s + 0.942·18-s − 2.75·19-s − 7.15·20-s + 4.36·21-s − 5.11·22-s − 1.25·23-s + 3.26·24-s + 25-s + 9.41·26-s + 0.384·27-s − 15.1·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\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 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{12} \cdot 5^{4} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$24.9885$$ Root analytic conductor: $$1.49526$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{280} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 2^{12} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## 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$C_2$ $$( 1 + p T + p T^{2} )^{2}$$
5$C_2^2$ $$1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
7$C_2$ $$( 1 + 5 T + p T^{2} )^{2}$$
good3$D_4\times C_2$ $$1 + 2 T + 5 T^{2} + 10 T^{3} + 16 T^{4} + 10 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
11$D_4\times C_2$ $$1 - 6 T + 36 T^{2} - 144 T^{3} + 587 T^{4} - 144 p T^{5} + 36 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2^2$ $$( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
17$C_2^2$$\times$$C_2^2$ $$( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} )$$
19$C_2$ $$( 1 - T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2}$$
23$D_4\times C_2$ $$1 + 6 T + 45 T^{2} + 246 T^{3} + 1208 T^{4} + 246 p T^{5} + 45 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
29$D_{4}$ $$( 1 - 2 T + 47 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 + 18 T + 172 T^{2} + 1152 T^{3} + 6483 T^{4} + 1152 p T^{5} + 172 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8}$$
37$D_4\times C_2$ $$1 + 12 T + 36 T^{2} - 12 p T^{3} - 4777 T^{4} - 12 p^{2} T^{5} + 36 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
41$D_4\times C_2$ $$1 - 122 T^{2} + 6651 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8}$$
43$D_4\times C_2$ $$1 + 6 T + 18 T^{2} + 276 T^{3} + 4223 T^{4} + 276 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 - 6 T + 90 T^{2} - 672 T^{3} + 5159 T^{4} - 672 p T^{5} + 90 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 + 12 T + 180 T^{2} + 1500 T^{3} + 13751 T^{4} + 1500 p T^{5} + 180 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 + 24 T + 6 p T^{2} + 3888 T^{3} + 34091 T^{4} + 3888 p T^{5} + 6 p^{3} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 - 6 T - 47 T^{2} + 234 T^{3} + 972 T^{4} + 234 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
67$D_4\times C_2$ $$1 + 18 T + 117 T^{2} - 6 T^{3} - 5008 T^{4} - 6 p T^{5} + 117 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8}$$
71$D_{4}$ $$( 1 - 24 T + 274 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 + 22 T + 290 T^{2} + 2736 T^{3} + 24239 T^{4} + 2736 p T^{5} + 290 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8}$$
79$D_4\times C_2$ $$1 + 24 T + 362 T^{2} + 4080 T^{3} + 37827 T^{4} + 4080 p T^{5} + 362 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8}$$
83$D_4\times C_2$ $$1 - 6 T + 18 T^{2} - 84 T^{3} - 4369 T^{4} - 84 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
89$D_4\times C_2$ $$1 + 24 T + 281 T^{2} + 2808 T^{3} + 27840 T^{4} + 2808 p T^{5} + 281 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8}$$
97$D_4\times C_2$ $$1 - 32 T + 512 T^{2} - 7008 T^{3} + 81038 T^{4} - 7008 p T^{5} + 512 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$