# Properties

 Label 8-384e4-1.1-c6e4-0-1 Degree $8$ Conductor $21743271936$ Sign $1$ Analytic cond. $6.09038\times 10^{7}$ Root an. cond. $9.39897$ Motivic weight $6$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 486·9-s − 1.15e4·17-s − 1.43e4·25-s + 7.49e4·41-s + 3.24e5·49-s + 2.51e6·73-s + 1.77e5·81-s − 5.50e6·89-s + 1.95e6·97-s − 6.32e6·113-s − 1.13e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 5.63e6·153-s + 157-s + 163-s + 167-s + 2.71e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 + 2/3·9-s − 2.35·17-s − 0.917·25-s + 1.08·41-s + 2.75·49-s + 6.45·73-s + 1/3·81-s − 7.80·89-s + 2.14·97-s − 4.38·113-s − 0.640·121-s − 1.57·153-s + 0.563·169-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(7-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{28} \cdot 3^{4}$$ Sign: $1$ Analytic conductor: $$6.09038\times 10^{7}$$ Root analytic conductor: $$9.39897$$ Motivic weight: $$6$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{384} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 3, 3, 3, 3 ),\ 1 )$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$2.325539905$$ $$L(\frac12)$$ $$\approx$$ $$2.325539905$$ $$L(4)$$ 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$C_2$ $$( 1 - p^{5} T^{2} )^{2}$$
good5$C_2^2$ $$( 1 + 7166 T^{2} + p^{12} T^{4} )^{2}$$
7$C_2^2$ $$( 1 - 162290 T^{2} + p^{12} T^{4} )^{2}$$
11$C_2^2$ $$( 1 + 567074 T^{2} + p^{12} T^{4} )^{2}$$
13$C_2^2$ $$( 1 - 1359218 T^{2} + p^{12} T^{4} )^{2}$$
17$C_2$ $$( 1 + 2898 T + p^{6} T^{2} )^{4}$$
19$C_2^2$ $$( 1 - 1743550 T^{2} + p^{12} T^{4} )^{2}$$
23$C_2^2$ $$( 1 - 293443490 T^{2} + p^{12} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 976603426 T^{2} + p^{12} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 1727677010 T^{2} + p^{12} T^{4} )^{2}$$
37$C_2^2$ $$( 1 - 4983392594 T^{2} + p^{12} T^{4} )^{2}$$
41$C_2$ $$( 1 - 18738 T + p^{6} T^{2} )^{4}$$
43$C_2^2$ $$( 1 - 5648607070 T^{2} + p^{12} T^{4} )^{2}$$
47$C_2^2$ $$( 1 + 1324300030 T^{2} + p^{12} T^{4} )^{2}$$
53$C_2^2$ $$( 1 - 23524698562 T^{2} + p^{12} T^{4} )^{2}$$
59$C_2^2$ $$( 1 - 26964818206 T^{2} + p^{12} T^{4} )^{2}$$
61$C_2^2$ $$( 1 + 8136149902 T^{2} + p^{12} T^{4} )^{2}$$
67$C_2^2$ $$( 1 + 78875702786 T^{2} + p^{12} T^{4} )^{2}$$
71$C_2^2$ $$( 1 - 79263457634 T^{2} + p^{12} T^{4} )^{2}$$
73$C_2$ $$( 1 - 8606 p T + p^{6} T^{2} )^{4}$$
79$C_2^2$ $$( 1 - 400981984274 T^{2} + p^{12} T^{4} )^{2}$$
83$C_2^2$ $$( 1 + 526419436610 T^{2} + p^{12} T^{4} )^{2}$$
89$C_2$ $$( 1 + 1375074 T + p^{6} T^{2} )^{4}$$
97$C_2$ $$( 1 - 489710 T + p^{6} T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$