# Properties

 Label 8-570e4-1.1-c5e4-0-5 Degree $8$ Conductor $105560010000$ Sign $1$ Analytic cond. $6.98460\times 10^{7}$ Root an. cond. $9.56131$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16·2-s − 36·3-s + 160·4-s + 100·5-s + 576·6-s − 108·7-s − 1.28e3·8-s + 810·9-s − 1.60e3·10-s − 246·11-s − 5.76e3·12-s + 640·13-s + 1.72e3·14-s − 3.60e3·15-s + 8.96e3·16-s − 612·17-s − 1.29e4·18-s + 1.44e3·19-s + 1.60e4·20-s + 3.88e3·21-s + 3.93e3·22-s − 1.24e3·23-s + 4.60e4·24-s + 6.25e3·25-s − 1.02e4·26-s − 1.45e4·27-s − 1.72e4·28-s + ⋯
 L(s)  = 1 − 2.82·2-s − 2.30·3-s + 5·4-s + 1.78·5-s + 6.53·6-s − 0.833·7-s − 7.07·8-s + 10/3·9-s − 5.05·10-s − 0.612·11-s − 11.5·12-s + 1.05·13-s + 2.35·14-s − 4.13·15-s + 35/4·16-s − 0.513·17-s − 9.42·18-s + 0.917·19-s + 8.94·20-s + 1.92·21-s + 1.73·22-s − 0.489·23-s + 16.3·24-s + 2·25-s − 2.97·26-s − 3.84·27-s − 4.16·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}$$ Sign: $1$ Analytic conductor: $$6.98460\times 10^{7}$$ Root analytic conductor: $$9.56131$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{570} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ 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)$
bad2$C_1$ $$( 1 + p^{2} T )^{4}$$
3$C_1$ $$( 1 + p^{2} T )^{4}$$
5$C_1$ $$( 1 - p^{2} T )^{4}$$
19$C_1$ $$( 1 - p^{2} T )^{4}$$
good7$C_2 \wr S_4$ $$1 + 108 T + 23900 T^{2} + 1713816 T^{3} + 552560598 T^{4} + 1713816 p^{5} T^{5} + 23900 p^{10} T^{6} + 108 p^{15} T^{7} + p^{20} T^{8}$$
11$C_2 \wr S_4$ $$1 + 246 T + 421460 T^{2} + 80210594 T^{3} + 8369673858 p T^{4} + 80210594 p^{5} T^{5} + 421460 p^{10} T^{6} + 246 p^{15} T^{7} + p^{20} T^{8}$$
13$C_2 \wr S_4$ $$1 - 640 T + 1056956 T^{2} - 361529700 T^{3} + 459852850718 T^{4} - 361529700 p^{5} T^{5} + 1056956 p^{10} T^{6} - 640 p^{15} T^{7} + p^{20} T^{8}$$
17$C_2 \wr S_4$ $$1 + 36 p T + 2493916 T^{2} + 3830078604 T^{3} + 2929546062758 T^{4} + 3830078604 p^{5} T^{5} + 2493916 p^{10} T^{6} + 36 p^{16} T^{7} + p^{20} T^{8}$$
23$C_2 \wr S_4$ $$1 + 54 p T + 15057544 T^{2} + 24084920250 T^{3} + 127312782620462 T^{4} + 24084920250 p^{5} T^{5} + 15057544 p^{10} T^{6} + 54 p^{16} T^{7} + p^{20} T^{8}$$
29$C_2 \wr S_4$ $$1 + 6230 T + 50227760 T^{2} + 162265143630 T^{3} + 1044362998531878 T^{4} + 162265143630 p^{5} T^{5} + 50227760 p^{10} T^{6} + 6230 p^{15} T^{7} + p^{20} T^{8}$$
31$C_2 \wr S_4$ $$1 + 11360 T + 97774932 T^{2} + 652434470416 T^{3} + 4089333295652182 T^{4} + 652434470416 p^{5} T^{5} + 97774932 p^{10} T^{6} + 11360 p^{15} T^{7} + p^{20} T^{8}$$
37$C_2 \wr S_4$ $$1 + 4792 T + 185999500 T^{2} + 783131830508 T^{3} + 16095708498883886 T^{4} + 783131830508 p^{5} T^{5} + 185999500 p^{10} T^{6} + 4792 p^{15} T^{7} + p^{20} T^{8}$$
41$C_2 \wr S_4$ $$1 - 9170 T + 343922232 T^{2} - 3465660766194 T^{3} + 52702052997241462 T^{4} - 3465660766194 p^{5} T^{5} + 343922232 p^{10} T^{6} - 9170 p^{15} T^{7} + p^{20} T^{8}$$
43$C_2 \wr S_4$ $$1 + 11412 T + 49864308 T^{2} - 2182228289304 T^{3} - 40366792660933082 T^{4} - 2182228289304 p^{5} T^{5} + 49864308 p^{10} T^{6} + 11412 p^{15} T^{7} + p^{20} T^{8}$$
47$C_2 \wr S_4$ $$1 + 29858 T + 908425208 T^{2} + 16046864111154 T^{3} + 297180955468482414 T^{4} + 16046864111154 p^{5} T^{5} + 908425208 p^{10} T^{6} + 29858 p^{15} T^{7} + p^{20} T^{8}$$
53$C_2 \wr S_4$ $$1 - 27498 T + 1596808876 T^{2} - 33463754731734 T^{3} + 988371145337755238 T^{4} - 33463754731734 p^{5} T^{5} + 1596808876 p^{10} T^{6} - 27498 p^{15} T^{7} + p^{20} T^{8}$$
59$C_2 \wr S_4$ $$1 - 54984 T + 3855461692 T^{2} - 124081177325352 T^{3} + 4493870636968603574 T^{4} - 124081177325352 p^{5} T^{5} + 3855461692 p^{10} T^{6} - 54984 p^{15} T^{7} + p^{20} T^{8}$$
61$C_2 \wr S_4$ $$1 - 20868 T + 1289735044 T^{2} - 39241717834460 T^{3} + 1483118740903875366 T^{4} - 39241717834460 p^{5} T^{5} + 1289735044 p^{10} T^{6} - 20868 p^{15} T^{7} + p^{20} T^{8}$$
67$C_2 \wr S_4$ $$1 + 20244 T + 3281754252 T^{2} + 94157149007604 T^{3} + 5199994947231751030 T^{4} + 94157149007604 p^{5} T^{5} + 3281754252 p^{10} T^{6} + 20244 p^{15} T^{7} + p^{20} T^{8}$$
71$C_2 \wr S_4$ $$1 - 86864 T + 9630749420 T^{2} - 493397548429296 T^{3} + 28310999748453748998 T^{4} - 493397548429296 p^{5} T^{5} + 9630749420 p^{10} T^{6} - 86864 p^{15} T^{7} + p^{20} T^{8}$$
73$C_2 \wr S_4$ $$1 + 3728 T + 928425484 T^{2} - 26462183539664 T^{3} + 6418662774571597190 T^{4} - 26462183539664 p^{5} T^{5} + 928425484 p^{10} T^{6} + 3728 p^{15} T^{7} + p^{20} T^{8}$$
79$C_2 \wr S_4$ $$1 - 164192 T + 15643001596 T^{2} - 955265739517024 T^{3} + 54103780177126731206 T^{4} - 955265739517024 p^{5} T^{5} + 15643001596 p^{10} T^{6} - 164192 p^{15} T^{7} + p^{20} T^{8}$$
83$C_2 \wr S_4$ $$1 + 60506 T + 8520751864 T^{2} + 352382473085458 T^{3} + 32730072408805508798 T^{4} + 352382473085458 p^{5} T^{5} + 8520751864 p^{10} T^{6} + 60506 p^{15} T^{7} + p^{20} T^{8}$$
89$C_2 \wr S_4$ $$1 - 113798 T + 14967753120 T^{2} - 1344398093521998 T^{3} +$$$$12\!\cdots\!98$$$$T^{4} - 1344398093521998 p^{5} T^{5} + 14967753120 p^{10} T^{6} - 113798 p^{15} T^{7} + p^{20} T^{8}$$
97$C_2 \wr S_4$ $$1 - 79440 T + 33692357804 T^{2} - 1965290512820340 T^{3} +$$$$43\!\cdots\!58$$$$T^{4} - 1965290512820340 p^{5} T^{5} + 33692357804 p^{10} T^{6} - 79440 p^{15} T^{7} + p^{20} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$