# Properties

 Degree $8$ Conductor $203928109056$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·3-s + 8·9-s + 8·11-s + 8·13-s − 16·23-s + 8·25-s + 12·27-s + 32·33-s + 8·37-s + 32·39-s − 16·47-s − 2·49-s − 8·59-s + 40·61-s − 64·69-s − 24·71-s − 24·73-s + 32·75-s + 23·81-s + 40·83-s − 24·97-s + 64·99-s + 24·107-s + 24·109-s + 32·111-s + 64·117-s − 4·121-s + ⋯
 L(s)  = 1 + 2.30·3-s + 8/3·9-s + 2.41·11-s + 2.21·13-s − 3.33·23-s + 8/5·25-s + 2.30·27-s + 5.57·33-s + 1.31·37-s + 5.12·39-s − 2.33·47-s − 2/7·49-s − 1.04·59-s + 5.12·61-s − 7.70·69-s − 2.84·71-s − 2.80·73-s + 3.69·75-s + 23/9·81-s + 4.39·83-s − 2.43·97-s + 6.43·99-s + 2.32·107-s + 2.29·109-s + 3.03·111-s + 5.91·117-s − 0.363·121-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{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^{20} \cdot 3^{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: $$2^{20} \cdot 3^{4} \cdot 7^{4}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{672} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{20} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$10.5150$$ $$L(\frac12)$$ $$\approx$$ $$10.5150$$ $$L(\frac{3}{2})$$ 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 $$1$$
3$C_2^2$ $$1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
7$C_2$ $$( 1 + T^{2} )^{2}$$
good5$D_4\times C_2$ $$1 - 8 T^{2} + 34 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8}$$
11$C_2$ $$( 1 - 2 T + p T^{2} )^{4}$$
13$D_{4}$ $$( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 - 44 T^{2} + 934 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8}$$
19$D_4\times C_2$ $$1 - 32 T^{2} + 690 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8}$$
23$C_2$ $$( 1 + 4 T + p T^{2} )^{4}$$
29$D_4\times C_2$ $$1 - 92 T^{2} + 3670 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8}$$
31$D_4\times C_2$ $$1 - 76 T^{2} + 2854 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8}$$
37$D_{4}$ $$( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2$ $$( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2}$$
43$D_4\times C_2$ $$1 - 20 T^{2} + 2646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}$$
47$D_{4}$ $$( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
53$D_4\times C_2$ $$1 + 52 T^{2} + 4246 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8}$$
59$D_{4}$ $$( 1 + 4 T + 104 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
61$D_{4}$ $$( 1 - 20 T + 204 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2}$$
67$D_4\times C_2$ $$1 - 4 T^{2} + 6934 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8}$$
71$D_{4}$ $$( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_{4}$ $$( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
79$C_2^2$ $$( 1 - 142 T^{2} + p^{2} T^{4} )^{2}$$
83$D_{4}$ $$( 1 - 20 T + 264 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 - 284 T^{2} + 35494 T^{4} - 284 p^{2} T^{6} + p^{4} T^{8}$$
97$D_{4}$ $$( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$