# Properties

 Label 8-882e4-1.1-c3e4-0-1 Degree $8$ Conductor $605165749776$ Sign $1$ Analytic cond. $7.33396\times 10^{6}$ Root an. cond. $7.21385$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·2-s + 4·4-s + 16·8-s + 80·11-s − 64·16-s − 320·22-s + 136·23-s + 200·25-s − 440·29-s + 64·32-s + 40·37-s − 1.36e3·43-s + 320·44-s − 544·46-s − 800·50-s + 1.25e3·53-s + 1.76e3·58-s + 192·64-s − 1.08e3·67-s + 1.68e3·71-s − 160·74-s + 1.52e3·79-s + 5.44e3·86-s + 1.28e3·88-s + 544·92-s + 800·100-s − 5.02e3·106-s + ⋯
 L(s)  = 1 − 1.41·2-s + 1/2·4-s + 0.707·8-s + 2.19·11-s − 16-s − 3.10·22-s + 1.23·23-s + 8/5·25-s − 2.81·29-s + 0.353·32-s + 0.177·37-s − 4.82·43-s + 1.09·44-s − 1.74·46-s − 2.26·50-s + 3.25·53-s + 3.98·58-s + 3/8·64-s − 1.96·67-s + 2.80·71-s − 0.251·74-s + 2.16·79-s + 6.82·86-s + 1.55·88-s + 0.616·92-s + 4/5·100-s − 4.60·106-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{8} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$7.33396\times 10^{6}$$ Root analytic conductor: $$7.21385$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{882} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.01620683825$$ $$L(\frac12)$$ $$\approx$$ $$0.01620683825$$ $$L(\frac{5}{2})$$ 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$C_2$ $$( 1 + p T + p^{2} T^{2} )^{2}$$
3 $$1$$
7 $$1$$
good5$C_2^3$ $$1 - 8 p^{2} T^{2} + 39 p^{4} T^{4} - 8 p^{8} T^{6} + p^{12} T^{8}$$
11$C_2^2$ $$( 1 - 40 T + 269 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
13$C_2^2$ $$( 1 + 344 T^{2} + p^{6} T^{4} )^{2}$$
17$C_2^3$ $$1 - 9824 T^{2} + 72373407 T^{4} - 9824 p^{6} T^{6} + p^{12} T^{8}$$
19$C_2^3$ $$1 - 13590 T^{2} + 137642219 T^{4} - 13590 p^{6} T^{6} + p^{12} T^{8}$$
23$C_2^2$ $$( 1 - 68 T - 7543 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
29$C_2$ $$( 1 + 110 T + p^{3} T^{2} )^{4}$$
31$C_2^3$ $$1 - 45470 T^{2} + 1180017219 T^{4} - 45470 p^{6} T^{6} + p^{12} T^{8}$$
37$C_2^2$ $$( 1 - 20 T - 50253 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
41$C_2^2$ $$( 1 + 135392 T^{2} + p^{6} T^{4} )^{2}$$
43$C_2$ $$( 1 + 340 T + p^{3} T^{2} )^{4}$$
47$C_2^3$ $$1 - 199454 T^{2} + 29002682787 T^{4} - 199454 p^{6} T^{6} + p^{12} T^{8}$$
53$C_2^2$ $$( 1 - 628 T + 245507 T^{2} - 628 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
59$C_2^3$ $$1 + 358042 T^{2} + 86013540123 T^{4} + 358042 p^{6} T^{6} + p^{12} T^{8}$$
61$C_2^3$ $$1 + 388440 T^{2} + 99365259239 T^{4} + 388440 p^{6} T^{6} + p^{12} T^{8}$$
67$C_2^2$ $$( 1 + 540 T - 9163 T^{2} + 540 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
71$C_2$ $$( 1 - 420 T + p^{3} T^{2} )^{4}$$
73$C_2^3$ $$1 - 693984 T^{2} + 330279565967 T^{4} - 693984 p^{6} T^{6} + p^{12} T^{8}$$
79$C_2^2$ $$( 1 - 760 T + 84561 T^{2} - 760 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
83$C_2^2$ $$( 1 + 251126 T^{2} + p^{6} T^{4} )^{2}$$
89$C_2^3$ $$1 - 81488 T^{2} - 490340996817 T^{4} - 81488 p^{6} T^{6} + p^{12} T^{8}$$
97$C_2^2$ $$( 1 + 1573296 T^{2} + p^{6} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$