# Properties

 Degree $8$ Conductor $1358954496$ Sign $1$ Motivic weight $2$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6·9-s + 24·13-s + 84·25-s + 152·37-s + 20·49-s + 88·61-s − 120·73-s − 45·81-s + 360·97-s − 552·109-s − 144·117-s + 52·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 316·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 − 2/3·9-s + 1.84·13-s + 3.35·25-s + 4.10·37-s + 0.408·49-s + 1.44·61-s − 1.64·73-s − 5/9·81-s + 3.71·97-s − 5.06·109-s − 1.23·117-s + 0.429·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.86·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \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^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{24} \cdot 3^{4}$$ Sign: $1$ Motivic weight: $$2$$ Character: induced by $\chi_{192} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{24} \cdot 3^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$3.64300$$ $$L(\frac12)$$ $$\approx$$ $$3.64300$$ $$L(2)$$ 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$C_2^2$ $$1 + 2 p T^{2} + p^{4} T^{4}$$
good5$C_2^2$ $$( 1 - 42 T^{2} + p^{4} T^{4} )^{2}$$
7$C_2^2$ $$( 1 - 10 T^{2} + p^{4} T^{4} )^{2}$$
11$C_2^2$ $$( 1 - 26 T^{2} + p^{4} T^{4} )^{2}$$
13$C_2$ $$( 1 - 6 T + p^{2} T^{2} )^{4}$$
17$C_2^2$ $$( 1 - 66 T^{2} + p^{4} T^{4} )^{2}$$
19$C_2^2$ $$( 1 + 614 T^{2} + p^{4} T^{4} )^{2}$$
23$C_2^2$ $$( 1 - 194 T^{2} + p^{4} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 714 T^{2} + p^{4} T^{4} )^{2}$$
31$C_2^2$ $$( 1 + 950 T^{2} + p^{4} T^{4} )^{2}$$
37$C_2$ $$( 1 - 38 T + p^{2} T^{2} )^{4}$$
41$C_2^2$ $$( 1 - 3330 T^{2} + p^{4} T^{4} )^{2}$$
43$C_2^2$ $$( 1 + 3590 T^{2} + p^{4} T^{4} )^{2}$$
47$C_2^2$ $$( 1 - 962 T^{2} + p^{4} T^{4} )^{2}$$
53$C_2^2$ $$( 1 - 5418 T^{2} + p^{4} T^{4} )^{2}$$
59$C_2^2$ $$( 1 - 6746 T^{2} + p^{4} T^{4} )^{2}$$
61$C_2$ $$( 1 - 22 T + p^{2} T^{2} )^{4}$$
67$C_2^2$ $$( 1 - 4090 T^{2} + p^{4} T^{4} )^{2}$$
71$C_2^2$ $$( 1 - 9218 T^{2} + p^{4} T^{4} )^{2}$$
73$C_2$ $$( 1 + 30 T + p^{2} T^{2} )^{4}$$
79$C_2^2$ $$( 1 + 11510 T^{2} + p^{4} T^{4} )^{2}$$
83$C_2^2$ $$( 1 - 8378 T^{2} + p^{4} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - 15810 T^{2} + p^{4} T^{4} )^{2}$$
97$C_2$ $$( 1 - 90 T + p^{2} T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$