# Properties

 Label 8-570e4-1.1-c7e4-0-0 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 − 1.49e3·7-s + 1.02e4·8-s + 7.29e3·9-s + 1.60e4·10-s − 2.91e3·11-s − 6.91e4·12-s − 3.69e3·13-s − 4.78e4·14-s − 5.40e4·15-s + 1.43e5·16-s + 16·17-s + 2.33e5·18-s + 2.74e4·19-s + 3.20e5·20-s + 1.61e5·21-s − 9.31e4·22-s − 7.30e4·23-s − 1.10e6·24-s + 1.56e5·25-s − 1.18e5·26-s − 3.93e5·27-s − 9.57e5·28-s + ⋯
 L(s)  = 1 + 2.82·2-s − 2.30·3-s + 5·4-s + 1.78·5-s − 6.53·6-s − 1.64·7-s + 7.07·8-s + 10/3·9-s + 5.05·10-s − 0.659·11-s − 11.5·12-s − 0.466·13-s − 4.66·14-s − 4.13·15-s + 35/4·16-s + 0.000789·17-s + 9.42·18-s + 0.917·19-s + 8.94·20-s + 3.80·21-s − 1.86·22-s − 1.25·23-s − 16.3·24-s + 2·25-s − 1.31·26-s − 3.84·27-s − 8.24·28-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$C_2 \wr S_4$ $$1 + 1496 T + 1348 p^{4} T^{2} + 3248754112 T^{3} + 4054349586902 T^{4} + 3248754112 p^{7} T^{5} + 1348 p^{18} T^{6} + 1496 p^{21} T^{7} + p^{28} T^{8}$$
11$C_2 \wr S_4$ $$1 + 2912 T + 6299692 p T^{2} + 170974862232 T^{3} + 176507609995122 p T^{4} + 170974862232 p^{7} T^{5} + 6299692 p^{15} T^{6} + 2912 p^{21} T^{7} + p^{28} T^{8}$$
13$C_2 \wr S_4$ $$1 + 3696 T + 184023652 T^{2} + 254871226936 T^{3} + 14516361784261494 T^{4} + 254871226936 p^{7} T^{5} + 184023652 p^{14} T^{6} + 3696 p^{21} T^{7} + p^{28} T^{8}$$
17$C_2 \wr S_4$ $$1 - 16 T + 857017292 T^{2} + 11057072782992 T^{3} + 325557069946951974 T^{4} + 11057072782992 p^{7} T^{5} + 857017292 p^{14} T^{6} - 16 p^{21} T^{7} + p^{28} T^{8}$$
23$C_2 \wr S_4$ $$1 + 73016 T + 10463536220 T^{2} + 652431450973848 T^{3} + 49932867735426482982 T^{4} + 652431450973848 p^{7} T^{5} + 10463536220 p^{14} T^{6} + 73016 p^{21} T^{7} + p^{28} T^{8}$$
29$C_2 \wr S_4$ $$1 + 137784 T + 42386981396 T^{2} + 2375840362140720 T^{3} +$$$$71\!\cdots\!34$$$$T^{4} + 2375840362140720 p^{7} T^{5} + 42386981396 p^{14} T^{6} + 137784 p^{21} T^{7} + p^{28} T^{8}$$
31$C_2 \wr S_4$ $$1 - 198072 T + 85820070508 T^{2} - 12336940811511128 T^{3} +$$$$33\!\cdots\!42$$$$T^{4} - 12336940811511128 p^{7} T^{5} + 85820070508 p^{14} T^{6} - 198072 p^{21} T^{7} + p^{28} T^{8}$$
37$C_2 \wr S_4$ $$1 - 207256 T + 152302694116 T^{2} - 26945217679675184 T^{3} +$$$$10\!\cdots\!22$$$$T^{4} - 26945217679675184 p^{7} T^{5} + 152302694116 p^{14} T^{6} - 207256 p^{21} T^{7} + p^{28} T^{8}$$
41$C_2 \wr S_4$ $$1 + 504056 T + 658336935524 T^{2} + 247661304371015808 T^{3} +$$$$17\!\cdots\!66$$$$T^{4} + 247661304371015808 p^{7} T^{5} + 658336935524 p^{14} T^{6} + 504056 p^{21} T^{7} + p^{28} T^{8}$$
43$C_2 \wr S_4$ $$1 + 250368 T + 794702741284 T^{2} + 114233542183566952 T^{3} +$$$$28\!\cdots\!38$$$$T^{4} + 114233542183566952 p^{7} T^{5} + 794702741284 p^{14} T^{6} + 250368 p^{21} T^{7} + p^{28} T^{8}$$
47$C_2 \wr S_4$ $$1 + 1000376 T + 2324322260924 T^{2} + 1544686466658023256 T^{3} +$$$$18\!\cdots\!06$$$$T^{4} + 1544686466658023256 p^{7} T^{5} + 2324322260924 p^{14} T^{6} + 1000376 p^{21} T^{7} + p^{28} T^{8}$$
53$C_2 \wr S_4$ $$1 + 2178688 T + 5028794631452 T^{2} + 7076462590603421760 T^{3} +$$$$90\!\cdots\!66$$$$T^{4} + 7076462590603421760 p^{7} T^{5} + 5028794631452 p^{14} T^{6} + 2178688 p^{21} T^{7} + p^{28} T^{8}$$
59$C_2 \wr S_4$ $$1 - 327976 T + 6413641153772 T^{2} + 1443811406273933784 T^{3} +$$$$18\!\cdots\!22$$$$T^{4} + 1443811406273933784 p^{7} T^{5} + 6413641153772 p^{14} T^{6} - 327976 p^{21} T^{7} + p^{28} T^{8}$$
61$C_2 \wr S_4$ $$1 - 572936 T + 1054077021484 T^{2} + 2799262298341413160 T^{3} -$$$$40\!\cdots\!98$$$$T^{4} + 2799262298341413160 p^{7} T^{5} + 1054077021484 p^{14} T^{6} - 572936 p^{21} T^{7} + p^{28} T^{8}$$
67$C_2 \wr S_4$ $$1 - 2017152 T + 20671485487084 T^{2} - 28219577873060887232 T^{3} +$$$$17\!\cdots\!06$$$$T^{4} - 28219577873060887232 p^{7} T^{5} + 20671485487084 p^{14} T^{6} - 2017152 p^{21} T^{7} + p^{28} T^{8}$$
71$C_2 \wr S_4$ $$1 - 2828960 T + 15546196545692 T^{2} - 59319424112529703200 T^{3} +$$$$19\!\cdots\!82$$$$T^{4} - 59319424112529703200 p^{7} T^{5} + 15546196545692 p^{14} T^{6} - 2828960 p^{21} T^{7} + p^{28} T^{8}$$
73$C_2 \wr S_4$ $$1 + 132392 T + 27176541756604 T^{2} - 7773597997820623208 T^{3} +$$$$36\!\cdots\!02$$$$T^{4} - 7773597997820623208 p^{7} T^{5} + 27176541756604 p^{14} T^{6} + 132392 p^{21} T^{7} + p^{28} T^{8}$$
79$C_2 \wr S_4$ $$1 - 3418408 T + 22454879188540 T^{2} -$$$$13\!\cdots\!52$$$$T^{3} +$$$$52\!\cdots\!90$$$$T^{4} -$$$$13\!\cdots\!52$$$$p^{7} T^{5} + 22454879188540 p^{14} T^{6} - 3418408 p^{21} T^{7} + p^{28} T^{8}$$
83$C_2 \wr S_4$ $$1 + 3201760 T + 54189243883340 T^{2} +$$$$25\!\cdots\!52$$$$T^{3} +$$$$16\!\cdots\!42$$$$T^{4} +$$$$25\!\cdots\!52$$$$p^{7} T^{5} + 54189243883340 p^{14} T^{6} + 3201760 p^{21} T^{7} + p^{28} T^{8}$$
89$C_2 \wr S_4$ $$1 + 1389392 T + 147421694160644 T^{2} +$$$$15\!\cdots\!32$$$$T^{3} +$$$$92\!\cdots\!10$$$$T^{4} +$$$$15\!\cdots\!32$$$$p^{7} T^{5} + 147421694160644 p^{14} T^{6} + 1389392 p^{21} T^{7} + p^{28} T^{8}$$
97$C_2 \wr S_4$ $$1 + 21061144 T + 447143586952804 T^{2} +$$$$52\!\cdots\!04$$$$T^{3} +$$$$58\!\cdots\!06$$$$T^{4} +$$$$52\!\cdots\!04$$$$p^{7} T^{5} + 447143586952804 p^{14} T^{6} + 21061144 p^{21} T^{7} + p^{28} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$