# Properties

 Label 8-39e8-1.1-c3e4-0-2 Degree $8$ Conductor $5.352\times 10^{12}$ Sign $1$ Analytic cond. $6.48606\times 10^{7}$ Root an. cond. $9.47322$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 9·4-s + 17·16-s + 96·17-s − 288·23-s − 240·25-s + 384·29-s + 1.28e3·43-s − 592·49-s − 984·53-s + 288·61-s − 153·64-s − 864·68-s + 4.32e3·79-s + 2.59e3·92-s + 2.16e3·100-s + 1.19e3·103-s + 2.49e3·107-s − 3.45e3·116-s − 2.17e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
 L(s)  = 1 − 9/8·4-s + 0.265·16-s + 1.36·17-s − 2.61·23-s − 1.91·25-s + 2.45·29-s + 4.56·43-s − 1.72·49-s − 2.55·53-s + 0.604·61-s − 0.298·64-s − 1.54·68-s + 6.15·79-s + 2.93·92-s + 2.15·100-s + 1.14·103-s + 2.25·107-s − 2.76·116-s − 1.63·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{8} \cdot 13^{8}$$ Sign: $1$ Analytic conductor: $$6.48606\times 10^{7}$$ Root analytic conductor: $$9.47322$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{8} \cdot 13^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$3.611995071$$ $$L(\frac12)$$ $$\approx$$ $$3.611995071$$ $$L(\frac{5}{2})$$ 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)$
bad3 $$1$$
13 $$1$$
good2$C_2^2 \wr C_2$ $$1 + 9 T^{2} + p^{6} T^{4} + 9 p^{6} T^{6} + p^{12} T^{8}$$
5$C_2^2 \wr C_2$ $$1 + 48 p T^{2} + 44302 T^{4} + 48 p^{7} T^{6} + p^{12} T^{8}$$
7$C_2^2 \wr C_2$ $$1 + 592 T^{2} + 310782 T^{4} + 592 p^{6} T^{6} + p^{12} T^{8}$$
11$C_2^2 \wr C_2$ $$1 + 2172 T^{2} + 3811270 T^{4} + 2172 p^{6} T^{6} + p^{12} T^{8}$$
17$D_{4}$ $$( 1 - 48 T - 1730 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
19$C_2^2 \wr C_2$ $$1 + 13552 T^{2} + 94862766 T^{4} + 13552 p^{6} T^{6} + p^{12} T^{8}$$
23$C_2$ $$( 1 + 72 T + p^{3} T^{2} )^{4}$$
29$D_{4}$ $$( 1 - 192 T + 45862 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
31$C_2^2 \wr C_2$ $$1 + 18592 T^{2} + 1722523710 T^{4} + 18592 p^{6} T^{6} + p^{12} T^{8}$$
37$C_2^2 \wr C_2$ $$1 + 92356 T^{2} + 6285341910 T^{4} + 92356 p^{6} T^{6} + p^{12} T^{8}$$
41$C_2^2 \wr C_2$ $$1 + 142320 T^{2} + 14548520830 T^{4} + 142320 p^{6} T^{6} + p^{12} T^{8}$$
43$D_{4}$ $$( 1 - 644 T + 250566 T^{2} - 644 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
47$C_2^2 \wr C_2$ $$1 + 385164 T^{2} + 58535996374 T^{4} + 385164 p^{6} T^{6} + p^{12} T^{8}$$
53$D_{4}$ $$( 1 + 492 T + 164158 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
59$C_2^2 \wr C_2$ $$1 + 758940 T^{2} + 227837028550 T^{4} + 758940 p^{6} T^{6} + p^{12} T^{8}$$
61$D_{4}$ $$( 1 - 144 T + 393094 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
67$C_2^2 \wr C_2$ $$1 + 611104 T^{2} + 187596510414 T^{4} + 611104 p^{6} T^{6} + p^{12} T^{8}$$
71$C_2^2 \wr C_2$ $$1 + 514188 T^{2} + 171350572726 T^{4} + 514188 p^{6} T^{6} + p^{12} T^{8}$$
73$C_2^2 \wr C_2$ $$1 + 659668 T^{2} + 389934034566 T^{4} + 659668 p^{6} T^{6} + p^{12} T^{8}$$
79$D_{4}$ $$( 1 - 2160 T + 2130910 T^{2} - 2160 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
83$C_2^2 \wr C_2$ $$1 + 823068 T^{2} + 604381635622 T^{4} + 823068 p^{6} T^{6} + p^{12} T^{8}$$
89$C_2^2 \wr C_2$ $$1 + 2258064 T^{2} + 2268670823038 T^{4} + 2258064 p^{6} T^{6} + p^{12} T^{8}$$
97$C_2^2 \wr C_2$ $$1 + 3513652 T^{2} + 4748412207846 T^{4} + 3513652 p^{6} T^{6} + p^{12} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$