# Properties

 Degree $8$ Conductor $21743271936$ Sign $1$ Motivic weight $2$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16·5-s + 2·9-s + 60·25-s − 208·29-s − 32·45-s − 36·49-s − 80·53-s − 200·73-s − 77·81-s + 200·97-s + 368·101-s − 444·121-s + 720·125-s + 127-s + 131-s + 137-s + 139-s + 3.32e3·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 36·169-s + 173-s + 179-s + 181-s + ⋯
 L(s)  = 1 − 3.19·5-s + 2/9·9-s + 12/5·25-s − 7.17·29-s − 0.711·45-s − 0.734·49-s − 1.50·53-s − 2.73·73-s − 0.950·81-s + 2.06·97-s + 3.64·101-s − 3.66·121-s + 5.75·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 22.9·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.213·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{28} \cdot 3^{4}$$ Sign: $1$ Motivic weight: $$2$$ Character: induced by $\chi_{384} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.0638087$$ $$L(\frac12)$$ $$\approx$$ $$0.0638087$$ $$L(2)$$ 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$C_2^2$ $$1 - 2 T^{2} + p^{4} T^{4}$$
good5$C_2$ $$( 1 + 4 T + p^{2} T^{2} )^{4}$$
7$C_2^2$ $$( 1 + 18 T^{2} + p^{4} T^{4} )^{2}$$
11$C_2^2$ $$( 1 + 222 T^{2} + p^{4} T^{4} )^{2}$$
13$C_2^2$ $$( 1 - 18 T^{2} + p^{4} T^{4} )^{2}$$
17$C_2^2$ $$( 1 - 258 T^{2} + p^{4} T^{4} )^{2}$$
19$C_2^2$ $$( 1 - 322 T^{2} + p^{4} T^{4} )^{2}$$
23$C_2^2$ $$( 1 - 802 T^{2} + p^{4} T^{4} )^{2}$$
29$C_2$ $$( 1 + 52 T + p^{2} T^{2} )^{4}$$
31$C_2^2$ $$( 1 + 1202 T^{2} + p^{4} T^{4} )^{2}$$
37$C_2^2$ $$( 1 + 142 T^{2} + p^{4} T^{4} )^{2}$$
41$C_2^2$ $$( 1 - 2082 T^{2} + p^{4} T^{4} )^{2}$$
43$C_2^2$ $$( 1 - 2402 T^{2} + p^{4} T^{4} )^{2}$$
47$C_2^2$ $$( 1 - 322 T^{2} + p^{4} T^{4} )^{2}$$
53$C_2$ $$( 1 + 20 T + p^{2} T^{2} )^{4}$$
59$C_2^2$ $$( 1 - 3618 T^{2} + p^{4} T^{4} )^{2}$$
61$C_2^2$ $$( 1 - 7122 T^{2} + p^{4} T^{4} )^{2}$$
67$C_2^2$ $$( 1 - 7042 T^{2} + p^{4} T^{4} )^{2}$$
71$C_2^2$ $$( 1 - 3682 T^{2} + p^{4} T^{4} )^{2}$$
73$C_2$ $$( 1 + 50 T + p^{2} T^{2} )^{4}$$
79$C_2^2$ $$( 1 + 6002 T^{2} + p^{4} T^{4} )^{2}$$
83$C_2^2$ $$( 1 + 3198 T^{2} + p^{4} T^{4} )^{2}$$
89$C_2^2$ $$( 1 + 10078 T^{2} + p^{4} T^{4} )^{2}$$
97$C_2$ $$( 1 - 50 T + p^{2} T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$