# Properties

 Label 8-1470e4-1.1-c1e4-0-1 Degree $8$ Conductor $4.669\times 10^{12}$ Sign $1$ Analytic cond. $18983.5$ Root an. cond. $3.42607$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4-s + 2·5-s + 9-s − 4·11-s + 12·19-s + 2·20-s + 5·25-s − 24·29-s − 4·31-s + 36-s − 8·41-s − 4·44-s + 2·45-s − 8·55-s + 16·59-s − 20·61-s − 64-s − 24·71-s + 12·76-s − 24·79-s + 20·89-s + 24·95-s − 4·99-s + 5·100-s + 20·101-s − 28·109-s − 24·116-s + ⋯
 L(s)  = 1 + 1/2·4-s + 0.894·5-s + 1/3·9-s − 1.20·11-s + 2.75·19-s + 0.447·20-s + 25-s − 4.45·29-s − 0.718·31-s + 1/6·36-s − 1.24·41-s − 0.603·44-s + 0.298·45-s − 1.07·55-s + 2.08·59-s − 2.56·61-s − 1/8·64-s − 2.84·71-s + 1.37·76-s − 2.70·79-s + 2.11·89-s + 2.46·95-s − 0.402·99-s + 1/2·100-s + 1.99·101-s − 2.68·109-s − 2.22·116-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: $1$ Analytic conductor: $$18983.5$$ Root analytic conductor: $$3.42607$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes 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

 $$L(1)$$ $$\approx$$ $$0.2050343942$$ $$L(\frac12)$$ $$\approx$$ $$0.2050343942$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4}$$
7 $$1$$
good11$C_2^2$ $$( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
13$C_2$ $$( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2}$$
17$C_2^3$ $$1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8}$$
19$C_2^2$ $$( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
23$C_2^3$ $$1 - 18 T^{2} - 205 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2$ $$( 1 + 6 T + p T^{2} )^{4}$$
31$C_2^2$ $$( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
37$C_2^3$ $$1 + 58 T^{2} + 1995 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8}$$
41$C_2$ $$( 1 + 2 T + p T^{2} )^{4}$$
43$C_2^2$ $$( 1 - 70 T^{2} + p^{2} T^{4} )^{2}$$
47$C_2^3$ $$1 + 30 T^{2} - 1309 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2^3$ $$1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8}$$
59$C_2^2$ $$( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
61$C_2^2$ $$( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^3$ $$1 + 70 T^{2} + 411 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8}$$
71$C_2$ $$( 1 + 6 T + p T^{2} )^{4}$$
73$C_2^3$ $$1 - 50 T^{2} - 2829 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8}$$
79$C_2^2$ $$( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
83$C_2^2$ $$( 1 - 102 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}$$
97$C_2^2$ $$( 1 - 94 T^{2} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$