# Properties

 Label 8-1800e4-1.1-c1e4-0-3 Degree $8$ Conductor $1.050\times 10^{13}$ Sign $1$ Analytic cond. $42677.4$ Root an. cond. $3.79118$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s + 2·4-s + 4·7-s − 4·8-s − 8·14-s + 8·16-s − 4·23-s + 8·28-s − 8·31-s − 8·32-s + 8·41-s + 8·46-s + 20·47-s − 12·49-s − 16·56-s + 16·62-s + 8·64-s − 8·71-s − 16·73-s − 32·79-s − 16·82-s − 8·89-s − 8·92-s − 40·94-s + 16·97-s + 24·98-s − 28·103-s + ⋯
 L(s)  = 1 − 1.41·2-s + 4-s + 1.51·7-s − 1.41·8-s − 2.13·14-s + 2·16-s − 0.834·23-s + 1.51·28-s − 1.43·31-s − 1.41·32-s + 1.24·41-s + 1.17·46-s + 2.91·47-s − 1.71·49-s − 2.13·56-s + 2.03·62-s + 64-s − 0.949·71-s − 1.87·73-s − 3.60·79-s − 1.76·82-s − 0.847·89-s − 0.834·92-s − 4.12·94-s + 1.62·97-s + 2.42·98-s − 2.75·103-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{12} \cdot 3^{8} \cdot 5^{8}$$ Sign: $1$ Analytic conductor: $$42677.4$$ Root analytic conductor: $$3.79118$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1800} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{12} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.5173448616$$ $$L(\frac12)$$ $$\approx$$ $$0.5173448616$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
3 $$1$$
5 $$1$$
good7$D_{4}$ $$( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
11$C_2^2$ $$( 1 - 18 T^{2} + p^{2} T^{4} )^{2}$$
13$C_2^2$ $$( 1 - 14 T^{2} + p^{2} T^{4} )^{2}$$
17$C_2^2$ $$( 1 + 22 T^{2} + p^{2} T^{4} )^{2}$$
19$D_4\times C_2$ $$1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}$$
23$D_{4}$ $$( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 10 T^{2} + p^{2} T^{4} )^{2}$$
31$D_{4}$ $$( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
37$C_2$ $$( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2}$$
41$D_{4}$ $$( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 - 68 T^{2} + 4266 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8}$$
47$D_{4}$ $$( 1 - 10 T + 116 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}$$
53$D_4\times C_2$ $$1 - 60 T^{2} + 3446 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 - 180 T^{2} + 14294 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 - 140 T^{2} + 11574 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8}$$
67$D_4\times C_2$ $$1 - 100 T^{2} + 10506 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8}$$
71$D_{4}$ $$( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_{4}$ $$( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
79$D_{4}$ $$( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 308 T^{2} + 37386 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8}$$
89$D_{4}$ $$( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
97$D_{4}$ $$( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$