# Properties

 Label 8-70e8-1.1-c1e4-0-5 Degree $8$ Conductor $5.765\times 10^{14}$ Sign $1$ Analytic cond. $2.34364\times 10^{6}$ Root an. cond. $6.25513$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 9-s + 6·11-s + 18·29-s − 48·71-s + 2·79-s + 9·81-s + 6·99-s + 22·109-s + 31·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 19·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
 L(s)  = 1 + 1/3·9-s + 1.80·11-s + 3.34·29-s − 5.69·71-s + 0.225·79-s + 81-s + 0.603·99-s + 2.10·109-s + 2.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.46·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \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^{8} \cdot 5^{8} \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: $$2^{8} \cdot 5^{8} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$2.34364\times 10^{6}$$ Root analytic conductor: $$6.25513$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{4900} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{8} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$8.557418768$$ $$L(\frac12)$$ $$\approx$$ $$8.557418768$$ $$L(\frac{3}{2})$$ 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 $$1$$
5 $$1$$
7 $$1$$
good3$C_2^3$ $$1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8}$$
11$C_2^2$ $$( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
13$C_2^3$ $$1 + 19 T^{2} + 192 T^{4} + 19 p^{2} T^{6} + p^{4} T^{8}$$
17$C_2^3$ $$1 - 29 T^{2} + 552 T^{4} - 29 p^{2} T^{6} + p^{4} T^{8}$$
19$C_2$ $$( 1 + p T^{2} )^{4}$$
23$C_2$ $$( 1 + p T^{2} )^{4}$$
29$C_2^2$ $$( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2}$$
31$C_2$ $$( 1 + p T^{2} )^{4}$$
37$C_2$ $$( 1 + p T^{2} )^{4}$$
41$C_2$ $$( 1 + p T^{2} )^{4}$$
43$C_2$ $$( 1 + p T^{2} )^{4}$$
47$C_2^3$ $$1 + 31 T^{2} - 1248 T^{4} + 31 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2$ $$( 1 + p T^{2} )^{4}$$
59$C_2$ $$( 1 + p T^{2} )^{4}$$
61$C_2$ $$( 1 + p T^{2} )^{4}$$
67$C_2$ $$( 1 + p T^{2} )^{4}$$
71$C_2$ $$( 1 + 12 T + p T^{2} )^{4}$$
73$C_2^2$ $$( 1 - 34 T^{2} + p^{2} T^{4} )^{2}$$
79$C_2^2$ $$( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} )^{2}$$
83$C_2^2$ $$( 1 + 86 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2$ $$( 1 + p T^{2} )^{4}$$
97$C_2^3$ $$1 - 149 T^{2} + 12792 T^{4} - 149 p^{2} T^{6} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$