# Properties

 Degree $8$ Conductor $4.669\times 10^{12}$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4-s + 4·5-s + 9-s + 10·11-s − 14·19-s + 4·20-s + 5·25-s − 12·31-s + 36-s + 36·41-s + 10·44-s + 4·45-s + 40·55-s − 8·59-s − 4·61-s − 64-s − 8·71-s − 14·76-s − 28·79-s − 20·89-s − 56·95-s + 10·99-s + 5·100-s + 16·101-s − 36·109-s + 47·121-s − 12·124-s + ⋯
 L(s)  = 1 + 1/2·4-s + 1.78·5-s + 1/3·9-s + 3.01·11-s − 3.21·19-s + 0.894·20-s + 25-s − 2.15·31-s + 1/6·36-s + 5.62·41-s + 1.50·44-s + 0.596·45-s + 5.39·55-s − 1.04·59-s − 0.512·61-s − 1/8·64-s − 0.949·71-s − 1.60·76-s − 3.15·79-s − 2.11·89-s − 5.74·95-s + 1.00·99-s + 1/2·100-s + 1.59·101-s − 3.44·109-s + 4.27·121-s − 1.07·124-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{1470} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$4.836592647$$ $$L(\frac12)$$ $$\approx$$ $$4.836592647$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$1 - T^{2} + T^{4}$$
3$C_2^2$ $$1 - T^{2} + T^{4}$$
5$C_2^2$ $$1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
7 $$1$$
good11$C_2^2$ $$( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2}$$
13$C_2^2$ $$( 1 - 25 T^{2} + p^{2} T^{4} )^{2}$$
17$C_2^2$$\times$$C_2^2$ $$( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} )$$
19$C_2$ $$( 1 - T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2}$$
23$C_2^3$ $$1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2$ $$( 1 + p T^{2} )^{4}$$
31$C_2^2$ $$( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
37$C_2^3$ $$1 + 49 T^{2} + 1032 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8}$$
41$C_2$ $$( 1 - 9 T + p T^{2} )^{4}$$
43$C_2^2$ $$( 1 + 14 T^{2} + p^{2} T^{4} )^{2}$$
47$C_2^3$ $$1 - 75 T^{2} + 3416 T^{4} - 75 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2^3$ $$1 + 105 T^{2} + 8216 T^{4} + 105 p^{2} T^{6} + p^{4} T^{8}$$
59$C_2^2$ $$( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
61$C_2^2$ $$( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^3$ $$1 + 98 T^{2} + 5115 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8}$$
71$C_2$ $$( 1 + 2 T + p T^{2} )^{4}$$
73$C_2^3$ $$1 + 130 T^{2} + 11571 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8}$$
79$C_2^2$ $$( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2}$$
83$C_2^2$ $$( 1 - 66 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}$$
97$C_2$ $$( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$