# Properties

 Label 8-1280e4-1.1-c1e4-0-14 Degree $8$ Conductor $2.684\times 10^{12}$ Sign $1$ Analytic cond. $10913.1$ Root an. cond. $3.19700$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·3-s + 8·9-s − 10·25-s − 20·27-s + 36·43-s + 12·67-s + 40·75-s + 50·81-s + 44·83-s + 24·89-s + 52·107-s + 44·121-s + 127-s − 144·129-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
 L(s)  = 1 − 2.30·3-s + 8/3·9-s − 2·25-s − 3.84·27-s + 5.48·43-s + 1.46·67-s + 4.61·75-s + 50/9·81-s + 4.82·83-s + 2.54·89-s + 5.02·107-s + 4·121-s + 0.0887·127-s − 12.6·129-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 − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{32} \cdot 5^{4}$$ Sign: $1$ Analytic conductor: $$10913.1$$ Root analytic conductor: $$3.19700$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1280} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{32} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.457780710$$ $$L(\frac12)$$ $$\approx$$ $$1.457780710$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 + p T^{2} )^{2}$$
good3$C_2^2$ $$( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
7$C_2^2$$\times$$C_2^2$ $$( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )$$
11$C_2$ $$( 1 - p T^{2} )^{4}$$
13$C_2$ $$( 1 + p T^{2} )^{4}$$
17$C_2$ $$( 1 - p T^{2} )^{4}$$
19$C_2$ $$( 1 - p T^{2} )^{4}$$
23$C_2^2$$\times$$C_2^2$ $$( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )$$
29$C_2^2$ $$( 1 - 22 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2$ $$( 1 + p T^{2} )^{4}$$
37$C_2$ $$( 1 + p T^{2} )^{4}$$
41$C_2^2$ $$( 1 + 62 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2^2$ $$( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2}$$
47$C_2^2$$\times$$C_2^2$ $$( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )$$
53$C_2$ $$( 1 + p T^{2} )^{4}$$
59$C_2$ $$( 1 - p T^{2} )^{4}$$
61$C_2$ $$( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2}$$
67$C_2^2$ $$( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_2$ $$( 1 + p T^{2} )^{4}$$
73$C_2$ $$( 1 - p T^{2} )^{4}$$
79$C_2$ $$( 1 + p T^{2} )^{4}$$
83$C_2^2$ $$( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2}$$
89$C_2$ $$( 1 - 6 T + p T^{2} )^{4}$$
97$C_2$ $$( 1 - p T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$