# Properties

 Label 8-882e4-1.1-c3e4-0-7 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 − 28·11-s − 64·16-s + 112·22-s + 280·23-s − 142·25-s + 1.14e3·29-s + 64·32-s + 76·37-s − 136·43-s − 112·44-s − 1.12e3·46-s + 568·50-s − 148·53-s − 4.57e3·58-s + 192·64-s − 1.36e3·67-s − 2.35e3·71-s − 304·74-s − 2.44e3·79-s + 544·86-s − 448·88-s + 1.12e3·92-s − 568·100-s + 592·106-s + ⋯
 L(s)  = 1 − 1.41·2-s + 1/2·4-s + 0.707·8-s − 0.767·11-s − 16-s + 1.08·22-s + 2.53·23-s − 1.13·25-s + 7.32·29-s + 0.353·32-s + 0.337·37-s − 0.482·43-s − 0.383·44-s − 3.58·46-s + 1.60·50-s − 0.383·53-s − 10.3·58-s + 3/8·64-s − 2.49·67-s − 3.93·71-s − 0.477·74-s − 3.47·79-s + 0.682·86-s − 0.542·88-s + 1.26·92-s − 0.567·100-s + 0.542·106-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

 $$L(2)$$ $$\approx$$ $$1.958764154$$ $$L(\frac12)$$ $$\approx$$ $$1.958764154$$ $$L(\frac{5}{2})$$ 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_2^2$$\times$$C_2^2$ $$( 1 - 18 T + 233 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} )( 1 + 18 T + 233 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} )$$
11$C_2^2$ $$( 1 + 14 T - 1135 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
13$C_2^2$ $$( 1 + 1802 T^{2} + p^{6} T^{4} )^{2}$$
17$C_2^3$ $$1 - 9824 T^{2} + 72373407 T^{4} - 9824 p^{6} T^{6} + p^{12} T^{8}$$
19$C_2^3$ $$1 - 13716 T^{2} + 141082775 T^{4} - 13716 p^{6} T^{6} + p^{12} T^{8}$$
23$C_2^2$ $$( 1 - 140 T + 7433 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
29$C_2$ $$( 1 - 286 T + p^{3} T^{2} )^{4}$$
31$C_2^3$ $$1 - 50870 T^{2} + 1700253219 T^{4} - 50870 p^{6} T^{6} + p^{12} T^{8}$$
37$C_2^2$ $$( 1 - 38 T - 49209 T^{2} - 38 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
41$C_2^2$ $$( 1 + 122000 T^{2} + p^{6} T^{4} )^{2}$$
43$C_2$ $$( 1 + 34 T + p^{3} T^{2} )^{4}$$
47$C_2^3$ $$1 + 66154 T^{2} - 6402863613 T^{4} + 66154 p^{6} T^{6} + p^{12} T^{8}$$
53$C_2^2$ $$( 1 + 74 T - 143401 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
59$C_2^3$ $$1 - 222260 T^{2} + 7218973959 T^{4} - 222260 p^{6} T^{6} + p^{12} T^{8}$$
61$C_2^3$ $$1 - 453762 T^{2} + 154379578283 T^{4} - 453762 p^{6} T^{6} + p^{12} T^{8}$$
67$C_2^2$ $$( 1 + 684 T + 167093 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
71$C_2$ $$( 1 + 588 T + p^{3} T^{2} )^{4}$$
73$C_2^3$ $$1 - 705072 T^{2} + 345792298895 T^{4} - 705072 p^{6} T^{6} + p^{12} T^{8}$$
79$C_2^2$ $$( 1 + 1220 T + 995361 T^{2} + 1220 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
83$C_2^2$ $$( 1 + 964772 T^{2} + p^{6} T^{4} )^{2}$$
89$C_2^3$ $$1 - 1028000 T^{2} + 559802709039 T^{4} - 1028000 p^{6} T^{6} + p^{12} T^{8}$$
97$C_2^2$ $$( 1 - 375456 T^{2} + p^{6} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$