# Properties

 Label 8-882e4-1.1-c2e4-0-15 Degree $8$ Conductor $605165749776$ Sign $1$ Analytic cond. $333591.$ Root an. cond. $4.90232$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·4-s + 36·11-s + 12·16-s + 60·23-s + 46·25-s − 48·29-s + 124·37-s − 8·43-s + 144·44-s − 156·53-s + 32·64-s + 116·67-s + 24·71-s − 220·79-s + 240·92-s + 184·100-s + 132·107-s − 140·109-s − 240·113-s − 192·116-s + 362·121-s + 127-s + 131-s + 137-s + 139-s + 496·148-s + 149-s + ⋯
 L(s)  = 1 + 4-s + 3.27·11-s + 3/4·16-s + 2.60·23-s + 1.83·25-s − 1.65·29-s + 3.35·37-s − 0.186·43-s + 3.27·44-s − 2.94·53-s + 1/2·64-s + 1.73·67-s + 0.338·71-s − 2.78·79-s + 2.60·92-s + 1.83·100-s + 1.23·107-s − 1.28·109-s − 2.12·113-s − 1.65·116-s + 2.99·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 3.35·148-s + 0.00671·149-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(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{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: $$333591.$$ Root analytic conductor: $$4.90232$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{882} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$10.99876730$$ $$L(\frac12)$$ $$\approx$$ $$10.99876730$$ $$L(2)$$ 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$C_2$ $$( 1 - p T^{2} )^{2}$$
3 $$1$$
7 $$1$$
good5$C_2^2$$\times$$C_2^2$ $$( 1 - 2 T - 21 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )( 1 + 2 T - 21 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )$$
11$D_{4}$ $$( 1 - 18 T + 305 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
13$D_4\times C_2$ $$1 - 412 T^{2} + 89190 T^{4} - 412 p^{4} T^{6} + p^{8} T^{8}$$
17$D_4\times C_2$ $$1 - 958 T^{2} + 1347 p^{2} T^{4} - 958 p^{4} T^{6} + p^{8} T^{8}$$
19$D_4\times C_2$ $$1 - 1426 T^{2} + 768939 T^{4} - 1426 p^{4} T^{6} + p^{8} T^{8}$$
23$D_{4}$ $$( 1 - 30 T + 1121 T^{2} - 30 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
29$D_{4}$ $$( 1 + 24 T + 1754 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 - 850 T^{2} + 1233867 T^{4} - 850 p^{4} T^{6} + p^{8} T^{8}$$
37$D_{4}$ $$( 1 - 62 T + 2547 T^{2} - 62 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
41$D_4\times C_2$ $$1 - 5500 T^{2} + 13185222 T^{4} - 5500 p^{4} T^{6} + p^{8} T^{8}$$
43$D_{4}$ $$( 1 + 4 T + 3630 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 - 3778 T^{2} + 13267131 T^{4} - 3778 p^{4} T^{6} + p^{8} T^{8}$$
53$D_{4}$ $$( 1 + 78 T + 6851 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
59$D_4\times C_2$ $$1 - 5410 T^{2} + 23946747 T^{4} - 5410 p^{4} T^{6} + p^{8} T^{8}$$
61$D_4\times C_2$ $$1 - 2302 T^{2} + 25403811 T^{4} - 2302 p^{4} T^{6} + p^{8} T^{8}$$
67$D_{4}$ $$( 1 - 58 T + 5769 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
71$D_{4}$ $$( 1 - 12 T + 8318 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 - 1390 T^{2} + 5504019 T^{4} - 1390 p^{4} T^{6} + p^{8} T^{8}$$
79$D_{4}$ $$( 1 + 110 T + 15057 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 + 380 T^{2} + 89625894 T^{4} + 380 p^{4} T^{6} + p^{8} T^{8}$$
89$D_4\times C_2$ $$1 - 958 T^{2} - 38888445 T^{4} - 958 p^{4} T^{6} + p^{8} T^{8}$$
97$D_4\times C_2$ $$1 - 26620 T^{2} + 330657414 T^{4} - 26620 p^{4} T^{6} + p^{8} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$