# Properties

 Label 8-570e4-1.1-c5e4-0-3 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 $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16·2-s + 36·3-s + 160·4-s + 100·5-s − 576·6-s + 268·7-s − 1.28e3·8-s + 810·9-s − 1.60e3·10-s + 338·11-s + 5.76e3·12-s + 1.30e3·13-s − 4.28e3·14-s + 3.60e3·15-s + 8.96e3·16-s + 542·17-s − 1.29e4·18-s + 1.44e3·19-s + 1.60e4·20-s + 9.64e3·21-s − 5.40e3·22-s − 88·23-s − 4.60e4·24-s + 6.25e3·25-s − 2.08e4·26-s + 1.45e4·27-s + 4.28e4·28-s + ⋯
 L(s)  = 1 − 2.82·2-s + 2.30·3-s + 5·4-s + 1.78·5-s − 6.53·6-s + 2.06·7-s − 7.07·8-s + 10/3·9-s − 5.05·10-s + 0.842·11-s + 11.5·12-s + 2.13·13-s − 5.84·14-s + 4.13·15-s + 35/4·16-s + 0.454·17-s − 9.42·18-s + 0.917·19-s + 8.94·20-s + 4.77·21-s − 2.38·22-s − 0.0346·23-s − 16.3·24-s + 2·25-s − 6.04·26-s + 3.84·27-s + 10.3·28-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: $$0$$ 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

 $$L(3)$$ $$\approx$$ $$39.06459287$$ $$L(\frac12)$$ $$\approx$$ $$39.06459287$$ $$L(\frac{7}{2})$$ 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$C_2 \wr S_4$ $$1 - 268 T + 50902 T^{2} - 8977064 T^{3} + 1428967202 T^{4} - 8977064 p^{5} T^{5} + 50902 p^{10} T^{6} - 268 p^{15} T^{7} + p^{20} T^{8}$$
11$C_2 \wr S_4$ $$1 - 338 T + 269922 T^{2} - 110307258 T^{3} + 66688722682 T^{4} - 110307258 p^{5} T^{5} + 269922 p^{10} T^{6} - 338 p^{15} T^{7} + p^{20} T^{8}$$
13$C_2 \wr S_4$ $$1 - 1302 T + 1269118 T^{2} - 895067978 T^{3} + 537434904594 T^{4} - 895067978 p^{5} T^{5} + 1269118 p^{10} T^{6} - 1302 p^{15} T^{7} + p^{20} T^{8}$$
17$C_2 \wr S_4$ $$1 - 542 T + 2520668 T^{2} - 53548602 p T^{3} + 281978762262 p T^{4} - 53548602 p^{6} T^{5} + 2520668 p^{10} T^{6} - 542 p^{15} T^{7} + p^{20} T^{8}$$
23$C_2 \wr S_4$ $$1 + 88 T + 7186400 T^{2} - 15291204024 T^{3} + 8716186394142 T^{4} - 15291204024 p^{5} T^{5} + 7186400 p^{10} T^{6} + 88 p^{15} T^{7} + p^{20} T^{8}$$
29$C_2 \wr S_4$ $$1 - 5014 T + 2124574 p T^{2} - 287580286770 T^{3} + 1700046374570514 T^{4} - 287580286770 p^{5} T^{5} + 2124574 p^{11} T^{6} - 5014 p^{15} T^{7} + p^{20} T^{8}$$
31$C_2 \wr S_4$ $$1 - 4962 T + 104254328 T^{2} - 407853713778 T^{3} + 4329315214967502 T^{4} - 407853713778 p^{5} T^{5} + 104254328 p^{10} T^{6} - 4962 p^{15} T^{7} + p^{20} T^{8}$$
37$C_2 \wr S_4$ $$1 - 8022 T + 6688862 p T^{2} - 1475777137002 T^{3} + 25079100324614082 T^{4} - 1475777137002 p^{5} T^{5} + 6688862 p^{11} T^{6} - 8022 p^{15} T^{7} + p^{20} T^{8}$$
41$C_2 \wr S_4$ $$1 - 2764 T + 249322694 T^{2} - 1632515334792 T^{3} + 31106725812996186 T^{4} - 1632515334792 p^{5} T^{5} + 249322694 p^{10} T^{6} - 2764 p^{15} T^{7} + p^{20} T^{8}$$
43$C_2 \wr S_4$ $$1 - 25346 T + 732962106 T^{2} - 11160511913746 T^{3} + 171470229081955978 T^{4} - 11160511913746 p^{5} T^{5} + 732962106 p^{10} T^{6} - 25346 p^{15} T^{7} + p^{20} T^{8}$$
47$C_2 \wr S_4$ $$1 - 10008 T + 648822656 T^{2} - 5238435651912 T^{3} + 210610538916792126 T^{4} - 5238435651912 p^{5} T^{5} + 648822656 p^{10} T^{6} - 10008 p^{15} T^{7} + p^{20} T^{8}$$
53$C_2 \wr S_4$ $$1 + 224 T + 665996288 T^{2} + 16576289045760 T^{3} + 137568305857212366 T^{4} + 16576289045760 p^{5} T^{5} + 665996288 p^{10} T^{6} + 224 p^{15} T^{7} + p^{20} T^{8}$$
59$C_2 \wr S_4$ $$1 - 29654 T + 2451079172 T^{2} - 53584799928294 T^{3} + 2525442592409067942 T^{4} - 53584799928294 p^{5} T^{5} + 2451079172 p^{10} T^{6} - 29654 p^{15} T^{7} + p^{20} T^{8}$$
61$C_2 \wr S_4$ $$1 - 27276 T + 3013858744 T^{2} - 69385767593540 T^{3} + 3668289795219064062 T^{4} - 69385767593540 p^{5} T^{5} + 3013858744 p^{10} T^{6} - 27276 p^{15} T^{7} + p^{20} T^{8}$$
67$C_2 \wr S_4$ $$1 - 26024 T + 4899369516 T^{2} - 88385416322536 T^{3} + 9477670181241697846 T^{4} - 88385416322536 p^{5} T^{5} + 4899369516 p^{10} T^{6} - 26024 p^{15} T^{7} + p^{20} T^{8}$$
71$C_2 \wr S_4$ $$1 + 26940 T + 4614821492 T^{2} + 115688848010220 T^{3} + 10068504844200664182 T^{4} + 115688848010220 p^{5} T^{5} + 4614821492 p^{10} T^{6} + 26940 p^{15} T^{7} + p^{20} T^{8}$$
73$C_2 \wr S_4$ $$1 + 60916 T + 6047709796 T^{2} + 339141266895724 T^{3} + 17251637000618592982 T^{4} + 339141266895724 p^{5} T^{5} + 6047709796 p^{10} T^{6} + 60916 p^{15} T^{7} + p^{20} T^{8}$$
79$C_2 \wr S_4$ $$1 - 34902 T + 11783798540 T^{2} - 303252461190558 T^{3} + 53617569850443048870 T^{4} - 303252461190558 p^{5} T^{5} + 11783798540 p^{10} T^{6} - 34902 p^{15} T^{7} + p^{20} T^{8}$$
83$C_2 \wr S_4$ $$1 + 48430 T + 8088541440 T^{2} + 430673209456374 T^{3} + 45339147155957777902 T^{4} + 430673209456374 p^{5} T^{5} + 8088541440 p^{10} T^{6} + 48430 p^{15} T^{7} + p^{20} T^{8}$$
89$C_2 \wr S_4$ $$1 + 38348 T + 17641780214 T^{2} + 533323445064888 T^{3} +$$$$14\!\cdots\!90$$$$T^{4} + 533323445064888 p^{5} T^{5} + 17641780214 p^{10} T^{6} + 38348 p^{15} T^{7} + p^{20} T^{8}$$
97$C_2 \wr S_4$ $$1 - 45942 T + 25565440046 T^{2} - 846184574898498 T^{3} +$$$$29\!\cdots\!66$$$$T^{4} - 846184574898498 p^{5} T^{5} + 25565440046 p^{10} T^{6} - 45942 p^{15} T^{7} + p^{20} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$