# Properties

 Label 8-147e4-1.1-c5e4-0-4 Degree $8$ Conductor $466948881$ Sign $1$ Analytic cond. $308966.$ Root an. cond. $4.85555$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3·2-s − 36·3-s − 25·4-s − 108·6-s − 75·8-s + 810·9-s + 402·11-s + 900·12-s − 462·13-s + 599·16-s − 276·17-s + 2.43e3·18-s − 510·19-s + 1.20e3·22-s + 6.90e3·23-s + 2.70e3·24-s − 4.84e3·25-s − 1.38e3·26-s − 1.45e4·27-s + 540·29-s + 6.41e3·31-s + 4.06e3·32-s − 1.44e4·33-s − 828·34-s − 2.02e4·36-s + 1.52e4·37-s − 1.53e3·38-s + ⋯
 L(s)  = 1 + 0.530·2-s − 2.30·3-s − 0.781·4-s − 1.22·6-s − 0.414·8-s + 10/3·9-s + 1.00·11-s + 1.80·12-s − 0.758·13-s + 0.584·16-s − 0.231·17-s + 1.76·18-s − 0.324·19-s + 0.531·22-s + 2.71·23-s + 0.956·24-s − 1.54·25-s − 0.402·26-s − 3.84·27-s + 0.119·29-s + 1.19·31-s + 0.701·32-s − 2.31·33-s − 0.122·34-s − 2.60·36-s + 1.83·37-s − 0.171·38-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{4} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$308966.$$ Root analytic conductor: $$4.85555$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{4} \cdot 7^{8} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$3.002754157$$ $$L(\frac12)$$ $$\approx$$ $$3.002754157$$ $$L(\frac{7}{2})$$ 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$C_1$ $$( 1 + p^{2} T )^{4}$$
7 $$1$$
good2$C_2 \wr S_4$ $$1 - 3 T + 17 p T^{2} - 51 p T^{3} + 83 p^{2} T^{4} - 51 p^{6} T^{5} + 17 p^{11} T^{6} - 3 p^{15} T^{7} + p^{20} T^{8}$$
5$C_2 \wr S_4$ $$1 + 4843 T^{2} - 12108 p^{2} T^{3} + 7549256 T^{4} - 12108 p^{7} T^{5} + 4843 p^{10} T^{6} + p^{20} T^{8}$$
11$C_2 \wr S_4$ $$1 - 402 T + 405841 T^{2} - 93416934 T^{3} + 77165216456 T^{4} - 93416934 p^{5} T^{5} + 405841 p^{10} T^{6} - 402 p^{15} T^{7} + p^{20} T^{8}$$
13$C_2 \wr S_4$ $$1 + 462 T + 336749 T^{2} - 500754 T^{3} + 123849672672 T^{4} - 500754 p^{5} T^{5} + 336749 p^{10} T^{6} + 462 p^{15} T^{7} + p^{20} T^{8}$$
17$C_2 \wr S_4$ $$1 + 276 T + 4084676 T^{2} + 280763388 T^{3} + 7517219673798 T^{4} + 280763388 p^{5} T^{5} + 4084676 p^{10} T^{6} + 276 p^{15} T^{7} + p^{20} T^{8}$$
19$C_2 \wr S_4$ $$1 + 510 T + 4007525 T^{2} + 2990607690 T^{3} + 14975063190924 T^{4} + 2990607690 p^{5} T^{5} + 4007525 p^{10} T^{6} + 510 p^{15} T^{7} + p^{20} T^{8}$$
23$C_2 \wr S_4$ $$1 - 300 p T + 32867228 T^{2} - 116109363780 T^{3} + 343244422915686 T^{4} - 116109363780 p^{5} T^{5} + 32867228 p^{10} T^{6} - 300 p^{16} T^{7} + p^{20} T^{8}$$
29$C_2 \wr S_4$ $$1 - 540 T + 29394199 T^{2} + 29299685892 T^{3} + 772430149366772 T^{4} + 29299685892 p^{5} T^{5} + 29394199 p^{10} T^{6} - 540 p^{15} T^{7} + p^{20} T^{8}$$
31$C_2 \wr S_4$ $$1 - 6410 T + 96802736 T^{2} - 491920044696 T^{3} + 3990219807562217 T^{4} - 491920044696 p^{5} T^{5} + 96802736 p^{10} T^{6} - 6410 p^{15} T^{7} + p^{20} T^{8}$$
37$C_2 \wr S_4$ $$1 - 15250 T + 230519621 T^{2} - 1675418532570 T^{3} + 17310190338885440 T^{4} - 1675418532570 p^{5} T^{5} + 230519621 p^{10} T^{6} - 15250 p^{15} T^{7} + p^{20} T^{8}$$
41$C_2 \wr S_4$ $$1 + 4308 T + 270683560 T^{2} + 2598901038204 T^{3} + 34018560333944366 T^{4} + 2598901038204 p^{5} T^{5} + 270683560 p^{10} T^{6} + 4308 p^{15} T^{7} + p^{20} T^{8}$$
43$C_2 \wr S_4$ $$1 - 29198 T + 787995265 T^{2} - 13076978932730 T^{3} + 187469286625894360 T^{4} - 13076978932730 p^{5} T^{5} + 787995265 p^{10} T^{6} - 29198 p^{15} T^{7} + p^{20} T^{8}$$
47$C_2 \wr S_4$ $$1 - 15060 T + 968029772 T^{2} - 10316869428852 T^{3} + 338556639544077654 T^{4} - 10316869428852 p^{5} T^{5} + 968029772 p^{10} T^{6} - 15060 p^{15} T^{7} + p^{20} T^{8}$$
53$C_2 \wr S_4$ $$1 - 13692 T + 1174874543 T^{2} - 9138918677148 T^{3} + 624374053991775828 T^{4} - 9138918677148 p^{5} T^{5} + 1174874543 p^{10} T^{6} - 13692 p^{15} T^{7} + p^{20} T^{8}$$
59$C_2 \wr S_4$ $$1 + 34830 T + 3047953637 T^{2} + 74240819598150 T^{3} + 3333299875817266188 T^{4} + 74240819598150 p^{5} T^{5} + 3047953637 p^{10} T^{6} + 34830 p^{15} T^{7} + p^{20} T^{8}$$
61$C_2 \wr S_4$ $$1 - 5364 T + 2845789028 T^{2} - 10211521644252 T^{3} + 3398342140250230278 T^{4} - 10211521644252 p^{5} T^{5} + 2845789028 p^{10} T^{6} - 5364 p^{15} T^{7} + p^{20} T^{8}$$
67$C_2 \wr S_4$ $$1 + 5994 T + 2554505537 T^{2} + 29219705413074 T^{3} + 3802215802346578704 T^{4} + 29219705413074 p^{5} T^{5} + 2554505537 p^{10} T^{6} + 5994 p^{15} T^{7} + p^{20} T^{8}$$
71$C_2 \wr S_4$ $$1 - 89268 T + 8738662172 T^{2} - 466802240277492 T^{3} + 25044546792022133910 T^{4} - 466802240277492 p^{5} T^{5} + 8738662172 p^{10} T^{6} - 89268 p^{15} T^{7} + p^{20} T^{8}$$
73$C_2 \wr S_4$ $$1 + 59638 T + 8655226553 T^{2} + 350574157952982 T^{3} + 27168426643986231860 T^{4} + 350574157952982 p^{5} T^{5} + 8655226553 p^{10} T^{6} + 59638 p^{15} T^{7} + p^{20} T^{8}$$
79$C_2 \wr S_4$ $$1 + 44062 T + 8526420688 T^{2} + 421600841789848 T^{3} + 33701233575740693881 T^{4} + 421600841789848 p^{5} T^{5} + 8526420688 p^{10} T^{6} + 44062 p^{15} T^{7} + p^{20} T^{8}$$
83$C_2 \wr S_4$ $$1 - 208446 T + 23363412401 T^{2} - 1724627286611514 T^{3} +$$$$11\!\cdots\!56$$$$T^{4} - 1724627286611514 p^{5} T^{5} + 23363412401 p^{10} T^{6} - 208446 p^{15} T^{7} + p^{20} T^{8}$$
89$C_2 \wr S_4$ $$1 - 77520 T + 5530024288 T^{2} + 567577230261264 T^{3} - 47925515398937104546 T^{4} + 567577230261264 p^{5} T^{5} + 5530024288 p^{10} T^{6} - 77520 p^{15} T^{7} + p^{20} T^{8}$$
97$C_2 \wr S_4$ $$1 - 188630 T + 43620869129 T^{2} - 4760435860011510 T^{3} +$$$$59\!\cdots\!64$$$$T^{4} - 4760435860011510 p^{5} T^{5} + 43620869129 p^{10} T^{6} - 188630 p^{15} T^{7} + p^{20} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$