# Properties

 Label 8-1200e4-1.1-c1e4-0-25 Degree $8$ Conductor $2.074\times 10^{12}$ Sign $1$ Analytic cond. $8430.11$ Root an. cond. $3.09548$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·3-s − 8·7-s + 8·9-s + 32·21-s − 12·27-s − 32·31-s − 32·37-s − 8·43-s + 32·49-s − 24·61-s − 64·63-s − 24·67-s − 32·73-s + 23·81-s + 128·93-s + 32·97-s − 24·103-s + 128·111-s − 20·121-s + 127-s + 32·129-s + 131-s + 137-s + 139-s − 128·147-s + 149-s + 151-s + ⋯
 L(s)  = 1 − 2.30·3-s − 3.02·7-s + 8/3·9-s + 6.98·21-s − 2.30·27-s − 5.74·31-s − 5.26·37-s − 1.21·43-s + 32/7·49-s − 3.07·61-s − 8.06·63-s − 2.93·67-s − 3.74·73-s + 23/9·81-s + 13.2·93-s + 3.24·97-s − 2.36·103-s + 12.1·111-s − 1.81·121-s + 0.0887·127-s + 2.81·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.5·147-s + 0.0819·149-s + 0.0813·151-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \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^{16} \cdot 3^{4} \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: $$2^{16} \cdot 3^{4} \cdot 5^{8}$$ Sign: $1$ Analytic conductor: $$8430.11$$ Root analytic conductor: $$3.09548$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1200} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 $$1$$
3$C_2^2$ $$1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
5 $$1$$
good7$C_2^2$ $$( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
11$C_2^2$ $$( 1 + 10 T^{2} + p^{2} T^{4} )^{2}$$
13$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
17$C_2^3$ $$1 - 254 T^{4} + p^{4} T^{8}$$
19$C_2^2$ $$( 1 - 22 T^{2} + p^{2} T^{4} )^{2}$$
23$C_2^3$ $$1 - 958 T^{4} + p^{4} T^{8}$$
29$C_2^2$ $$( 1 + 26 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2$ $$( 1 + 8 T + p T^{2} )^{4}$$
37$C_2^2$ $$( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2^2$ $$( 1 - 50 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2^2$ $$( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
47$C_2^3$ $$1 + 3682 T^{4} + p^{4} T^{8}$$
53$C_2^3$ $$1 - 3854 T^{4} + p^{4} T^{8}$$
59$C_2^2$ $$( 1 + 86 T^{2} + p^{2} T^{4} )^{2}$$
61$C_2$ $$( 1 + 6 T + p T^{2} )^{4}$$
67$C_2^2$ $$( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_2^2$ $$( 1 - 14 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2^2$ $$( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}$$
79$C_2$ $$( 1 - p T^{2} )^{4}$$
83$C_2^3$ $$1 - 12878 T^{4} + p^{4} T^{8}$$
89$C_2^2$ $$( 1 + 50 T^{2} + p^{2} T^{4} )^{2}$$
97$C_2^2$ $$( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$