# Properties

 Label 8-570e4-1.1-c7e4-0-2 Degree $8$ Conductor $105560010000$ Sign $1$ Analytic cond. $1.00521\times 10^{9}$ Root an. cond. $13.3438$ Motivic weight $7$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 32·2-s + 108·3-s + 640·4-s + 500·5-s + 3.45e3·6-s − 1.31e3·7-s + 1.02e4·8-s + 7.29e3·9-s + 1.60e4·10-s − 6.57e3·11-s + 6.91e4·12-s − 1.75e4·13-s − 4.21e4·14-s + 5.40e4·15-s + 1.43e5·16-s − 2.84e4·17-s + 2.33e5·18-s − 2.74e4·19-s + 3.20e5·20-s − 1.42e5·21-s − 2.10e5·22-s − 3.21e4·23-s + 1.10e6·24-s + 1.56e5·25-s − 5.62e5·26-s + 3.93e5·27-s − 8.42e5·28-s + ⋯
 L(s)  = 1 + 2.82·2-s + 2.30·3-s + 5·4-s + 1.78·5-s + 6.53·6-s − 1.45·7-s + 7.07·8-s + 10/3·9-s + 5.05·10-s − 1.48·11-s + 11.5·12-s − 2.21·13-s − 4.10·14-s + 4.13·15-s + 35/4·16-s − 1.40·17-s + 9.42·18-s − 0.917·19-s + 8.94·20-s − 3.34·21-s − 4.20·22-s − 0.550·23-s + 16.3·24-s + 2·25-s − 6.27·26-s + 3.84·27-s − 7.25·28-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(8-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+7/2)^{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: $$1.00521\times 10^{9}$$ Root analytic conductor: $$13.3438$$ Motivic weight: $$7$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{570} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )$$

## Particular Values

 $$L(4)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{9}{2})$$ 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$C_1$ $$( 1 - p^{3} T )^{4}$$
3$C_1$ $$( 1 - p^{3} T )^{4}$$
5$C_1$ $$( 1 - p^{3} T )^{4}$$
19$C_1$ $$( 1 + p^{3} T )^{4}$$
good7$C_2 \wr S_4$ $$1 + 188 p T + 3092668 T^{2} + 398881876 p T^{3} + 3643191635462 T^{4} + 398881876 p^{8} T^{5} + 3092668 p^{14} T^{6} + 188 p^{22} T^{7} + p^{28} T^{8}$$
11$C_2 \wr S_4$ $$1 + 6570 T + 55802072 T^{2} + 148633201962 T^{3} + 992291347332990 T^{4} + 148633201962 p^{7} T^{5} + 55802072 p^{14} T^{6} + 6570 p^{21} T^{7} + p^{28} T^{8}$$
13$C_2 \wr S_4$ $$1 + 17578 T + 221418772 T^{2} + 1876174025262 T^{3} + 15069896273646150 T^{4} + 1876174025262 p^{7} T^{5} + 221418772 p^{14} T^{6} + 17578 p^{21} T^{7} + p^{28} T^{8}$$
17$C_2 \wr S_4$ $$1 + 28466 T + 837051992 T^{2} + 8365939855630 T^{3} + 229175463710842190 T^{4} + 8365939855630 p^{7} T^{5} + 837051992 p^{14} T^{6} + 28466 p^{21} T^{7} + p^{28} T^{8}$$
23$C_2 \wr S_4$ $$1 + 32136 T + 7488127196 T^{2} + 510556935133352 T^{3} + 26186381560297811238 T^{4} + 510556935133352 p^{7} T^{5} + 7488127196 p^{14} T^{6} + 32136 p^{21} T^{7} + p^{28} T^{8}$$
29$C_2 \wr S_4$ $$1 + 159122 T + 64719167228 T^{2} + 6654469087470278 T^{3} +$$$$15\!\cdots\!62$$$$T^{4} + 6654469087470278 p^{7} T^{5} + 64719167228 p^{14} T^{6} + 159122 p^{21} T^{7} + p^{28} T^{8}$$
31$C_2 \wr S_4$ $$1 + 67974 T + 58226470540 T^{2} - 1806216343105522 T^{3} +$$$$14\!\cdots\!74$$$$T^{4} - 1806216343105522 p^{7} T^{5} + 58226470540 p^{14} T^{6} + 67974 p^{21} T^{7} + p^{28} T^{8}$$
37$C_2 \wr S_4$ $$1 + 823702 T + 490118548108 T^{2} + 216128024231822626 T^{3} +$$$$75\!\cdots\!18$$$$T^{4} + 216128024231822626 p^{7} T^{5} + 490118548108 p^{14} T^{6} + 823702 p^{21} T^{7} + p^{28} T^{8}$$
41$C_2 \wr S_4$ $$1 + 781924 T + 767506780388 T^{2} + 382623949748239252 T^{3} +$$$$22\!\cdots\!30$$$$T^{4} + 382623949748239252 p^{7} T^{5} + 767506780388 p^{14} T^{6} + 781924 p^{21} T^{7} + p^{28} T^{8}$$
43$C_2 \wr S_4$ $$1 + 1115638 T + 737266652992 T^{2} + 356238832104359286 T^{3} +$$$$19\!\cdots\!38$$$$T^{4} + 356238832104359286 p^{7} T^{5} + 737266652992 p^{14} T^{6} + 1115638 p^{21} T^{7} + p^{28} T^{8}$$
47$C_2 \wr S_4$ $$1 + 209160 T + 863688190268 T^{2} + 197548391176963560 T^{3} +$$$$56\!\cdots\!98$$$$T^{4} + 197548391176963560 p^{7} T^{5} + 863688190268 p^{14} T^{6} + 209160 p^{21} T^{7} + p^{28} T^{8}$$
53$C_2 \wr S_4$ $$1 + 16008 p T + 4377691198700 T^{2} + 2779662163460513976 T^{3} +$$$$75\!\cdots\!54$$$$T^{4} + 2779662163460513976 p^{7} T^{5} + 4377691198700 p^{14} T^{6} + 16008 p^{22} T^{7} + p^{28} T^{8}$$
59$C_2 \wr S_4$ $$1 + 3677830 T + 12144428527772 T^{2} + 27009966703230094102 T^{3} +$$$$48\!\cdots\!34$$$$T^{4} + 27009966703230094102 p^{7} T^{5} + 12144428527772 p^{14} T^{6} + 3677830 p^{21} T^{7} + p^{28} T^{8}$$
61$C_2 \wr S_4$ $$1 + 1161072 T + 7069720468828 T^{2} + 313029995829507664 T^{3} +$$$$19\!\cdots\!70$$$$T^{4} + 313029995829507664 p^{7} T^{5} + 7069720468828 p^{14} T^{6} + 1161072 p^{21} T^{7} + p^{28} T^{8}$$
67$C_2 \wr S_4$ $$1 + 6154740 T + 22793858868268 T^{2} + 76558374281921978036 T^{3} +$$$$21\!\cdots\!34$$$$T^{4} + 76558374281921978036 p^{7} T^{5} + 22793858868268 p^{14} T^{6} + 6154740 p^{21} T^{7} + p^{28} T^{8}$$
71$C_2 \wr S_4$ $$1 + 4456224 T + 35224215958364 T^{2} +$$$$11\!\cdots\!28$$$$T^{3} +$$$$47\!\cdots\!26$$$$T^{4} +$$$$11\!\cdots\!28$$$$p^{7} T^{5} + 35224215958364 p^{14} T^{6} + 4456224 p^{21} T^{7} + p^{28} T^{8}$$
73$C_2 \wr S_4$ $$1 - 1057792 T + 25051579433164 T^{2} - 23683837938073896192 T^{3} +$$$$38\!\cdots\!02$$$$T^{4} - 23683837938073896192 p^{7} T^{5} + 25051579433164 p^{14} T^{6} - 1057792 p^{21} T^{7} + p^{28} T^{8}$$
79$C_2 \wr S_4$ $$1 - 2910090 T + 76806351838396 T^{2} -$$$$16\!\cdots\!98$$$$T^{3} +$$$$22\!\cdots\!26$$$$T^{4} -$$$$16\!\cdots\!98$$$$p^{7} T^{5} + 76806351838396 p^{14} T^{6} - 2910090 p^{21} T^{7} + p^{28} T^{8}$$
83$C_2 \wr S_4$ $$1 + 1767198 T + 77723349910148 T^{2} +$$$$19\!\cdots\!66$$$$T^{3} +$$$$27\!\cdots\!98$$$$T^{4} +$$$$19\!\cdots\!66$$$$p^{7} T^{5} + 77723349910148 p^{14} T^{6} + 1767198 p^{21} T^{7} + p^{28} T^{8}$$
89$C_2 \wr S_4$ $$1 + 3677360 T + 91147101194132 T^{2} -$$$$13\!\cdots\!48$$$$T^{3} +$$$$30\!\cdots\!94$$$$T^{4} -$$$$13\!\cdots\!48$$$$p^{7} T^{5} + 91147101194132 p^{14} T^{6} + 3677360 p^{21} T^{7} + p^{28} T^{8}$$
97$C_2 \wr S_4$ $$1 + 10419094 T + 332866439705620 T^{2} +$$$$23\!\cdots\!54$$$$T^{3} +$$$$40\!\cdots\!70$$$$T^{4} +$$$$23\!\cdots\!54$$$$p^{7} T^{5} + 332866439705620 p^{14} T^{6} + 10419094 p^{21} T^{7} + p^{28} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$