# Properties

 Label 8-39e8-1.1-c3e4-0-0 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 + 13·4-s − 36·7-s + 57·16-s − 84·19-s + 80·25-s − 468·28-s + 604·31-s − 184·37-s + 880·43-s − 96·49-s + 656·61-s + 117·64-s − 3.05e3·67-s + 312·73-s − 1.09e3·76-s − 720·79-s + 344·97-s + 1.04e3·100-s + 3.39e3·103-s − 3.38e3·109-s − 2.05e3·112-s − 4.57e3·121-s + 7.85e3·124-s + 127-s + 131-s + 3.02e3·133-s + 137-s + ⋯
 L(s)  = 1 + 13/8·4-s − 1.94·7-s + 0.890·16-s − 1.01·19-s + 0.639·25-s − 3.15·28-s + 3.49·31-s − 0.817·37-s + 3.12·43-s − 0.279·49-s + 1.37·61-s + 0.228·64-s − 5.56·67-s + 0.500·73-s − 1.64·76-s − 1.02·79-s + 0.360·97-s + 1.03·100-s + 3.24·103-s − 2.97·109-s − 1.73·112-s − 3.43·121-s + 5.68·124-s + 0.000698·127-s + 0.000666·131-s + 1.97·133-s + 0.000623·137-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

 $$L(2)$$ $$\approx$$ $$0.6398019709$$ $$L(\frac12)$$ $$\approx$$ $$0.6398019709$$ $$L(\frac{5}{2})$$ 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$C_2^2 \wr C_2$ $$1 - 13 T^{2} + 7 p^{4} T^{4} - 13 p^{6} T^{6} + p^{12} T^{8}$$
5$C_2^2 \wr C_2$ $$1 - 16 p T^{2} + 24462 T^{4} - 16 p^{7} T^{6} + p^{12} T^{8}$$
7$D_{4}$ $$( 1 + 18 T + 534 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
11$C_2^2 \wr C_2$ $$1 + 4572 T^{2} + 8634710 T^{4} + 4572 p^{6} T^{6} + p^{12} T^{8}$$
17$C_2^2 \wr C_2$ $$1 + 16772 T^{2} + 118361542 T^{4} + 16772 p^{6} T^{6} + p^{12} T^{8}$$
19$D_{4}$ $$( 1 + 42 T + 2742 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
23$C_2^2 \wr C_2$ $$1 + 12 T^{2} - 393658 T^{4} + 12 p^{6} T^{6} + p^{12} T^{8}$$
29$C_2^2 \wr C_2$ $$1 + 22356 T^{2} - 27485674 T^{4} + 22356 p^{6} T^{6} + p^{12} T^{8}$$
31$D_{4}$ $$( 1 - 302 T + 2650 p T^{2} - 302 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
37$D_{4}$ $$( 1 + 92 T + 43774 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
41$C_2^2 \wr C_2$ $$1 + 227840 T^{2} + 22474730590 T^{4} + 227840 p^{6} T^{6} + p^{12} T^{8}$$
43$D_{4}$ $$( 1 - 440 T + 203686 T^{2} - 440 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
47$C_2^2 \wr C_2$ $$1 + 270652 T^{2} + 39420081222 T^{4} + 270652 p^{6} T^{6} + p^{12} T^{8}$$
53$C_2^2 \wr C_2$ $$1 + 438372 T^{2} + 89997219542 T^{4} + 438372 p^{6} T^{6} + p^{12} T^{8}$$
59$C_2^2 \wr C_2$ $$1 + 598556 T^{2} + 161876495254 T^{4} + 598556 p^{6} T^{6} + p^{12} T^{8}$$
61$D_{4}$ $$( 1 - 328 T + 368086 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
67$D_{4}$ $$( 1 + 1526 T + 1116358 T^{2} + 1526 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
71$C_2^2 \wr C_2$ $$1 + 500316 T^{2} + 133803927398 T^{4} + 500316 p^{6} T^{6} + p^{12} T^{8}$$
73$D_{4}$ $$( 1 - 156 T - 293274 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
79$D_{4}$ $$( 1 + 360 T + 287790 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
83$C_2^2 \wr C_2$ $$1 + 2272508 T^{2} + 1944939568822 T^{4} + 2272508 p^{6} T^{6} + p^{12} T^{8}$$
89$C_2^2 \wr C_2$ $$1 + 913024 T^{2} + 920290760478 T^{4} + 913024 p^{6} T^{6} + p^{12} T^{8}$$
97$D_{4}$ $$( 1 - 172 T + 1202710 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$