# Properties

 Label 8-882e4-1.1-c3e4-0-0 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 − 4·11-s − 64·16-s + 16·22-s − 60·23-s + 18·25-s + 848·29-s + 64·32-s − 492·37-s − 1.13e3·43-s − 16·44-s + 240·46-s − 72·50-s − 1.09e3·53-s − 3.39e3·58-s + 192·64-s − 1.30e3·67-s + 3.08e3·71-s + 1.96e3·74-s − 944·79-s + 4.54e3·86-s − 64·88-s − 240·92-s + 72·100-s + 4.38e3·106-s + ⋯
 L(s)  = 1 − 1.41·2-s + 1/2·4-s + 0.707·8-s − 0.109·11-s − 16-s + 0.155·22-s − 0.543·23-s + 0.143·25-s + 5.42·29-s + 0.353·32-s − 2.18·37-s − 4.02·43-s − 0.0548·44-s + 0.769·46-s − 0.203·50-s − 2.84·53-s − 7.67·58-s + 3/8·64-s − 2.37·67-s + 5.14·71-s + 3.09·74-s − 1.34·79-s + 5.69·86-s − 0.0775·88-s − 0.271·92-s + 0.0719·100-s + 4.01·106-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

 $$L(2)$$ $$\approx$$ $$0.002200279424$$ $$L(\frac12)$$ $$\approx$$ $$0.002200279424$$ $$L(\frac{5}{2})$$ 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$C_2^3$ $$1 - 18 T^{2} - 15301 T^{4} - 18 p^{6} T^{6} + p^{12} T^{8}$$
11$C_2^2$ $$( 1 + 2 T - 1327 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
13$C_2^2$ $$( 1 + 3466 T^{2} + p^{6} T^{4} )^{2}$$
17$C_2^3$ $$1 - 7738 T^{2} + 35739075 T^{4} - 7738 p^{6} T^{6} + p^{12} T^{8}$$
19$C_2^3$ $$1 + 9482 T^{2} + 42862443 T^{4} + 9482 p^{6} T^{6} + p^{12} T^{8}$$
23$C_2^2$ $$( 1 + 30 T - 11267 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
29$C_2$ $$( 1 - 212 T + p^{3} T^{2} )^{4}$$
31$C_2^3$ $$1 - 14110 T^{2} - 688411581 T^{4} - 14110 p^{6} T^{6} + p^{12} T^{8}$$
37$C_2^2$ $$( 1 + 246 T + 9863 T^{2} + 246 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
41$C_2^2$ $$( 1 + 35530 T^{2} + p^{6} T^{4} )^{2}$$
43$C_2$ $$( 1 + 284 T + p^{3} T^{2} )^{4}$$
47$C_2^3$ $$1 - 203934 T^{2} + 30809861027 T^{4} - 203934 p^{6} T^{6} + p^{12} T^{8}$$
53$C_2^2$ $$( 1 + 548 T + 151427 T^{2} + 548 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
59$C_2^3$ $$1 + 38394 T^{2} - 40706434405 T^{4} + 38394 p^{6} T^{6} + p^{12} T^{8}$$
61$C_2^3$ $$1 - 185770 T^{2} - 17009881461 T^{4} - 185770 p^{6} T^{6} + p^{12} T^{8}$$
67$C_2^2$ $$( 1 + 652 T + 124341 T^{2} + 652 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
71$C_2$ $$( 1 - 770 T + p^{3} T^{2} )^{4}$$
73$C_2^3$ $$1 + 172238 T^{2} - 121668297645 T^{4} + 172238 p^{6} T^{6} + p^{12} T^{8}$$
79$C_2^2$ $$( 1 + 472 T - 270255 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
83$C_2^2$ $$( 1 + 1110166 T^{2} + p^{6} T^{4} )^{2}$$
89$C_2^3$ $$1 - 897450 T^{2} + 308435211539 T^{4} - 897450 p^{6} T^{6} + p^{12} T^{8}$$
97$C_2^2$ $$( 1 + 1732546 T^{2} + p^{6} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$