# Properties

 Label 8-210e4-1.1-c11e4-0-0 Degree $8$ Conductor $1944810000$ Sign $1$ Analytic cond. $6.77794\times 10^{8}$ Root an. cond. $12.7024$ Motivic weight $11$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 128·2-s + 972·3-s + 1.02e4·4-s + 1.25e4·5-s + 1.24e5·6-s + 6.72e4·7-s + 6.55e5·8-s + 5.90e5·9-s + 1.60e6·10-s + 4.58e5·11-s + 9.95e6·12-s + 1.57e6·13-s + 8.60e6·14-s + 1.21e7·15-s + 3.67e7·16-s + 8.67e6·17-s + 7.55e7·18-s + 1.24e7·19-s + 1.28e8·20-s + 6.53e7·21-s + 5.86e7·22-s + 5.13e5·23-s + 6.37e8·24-s + 9.76e7·25-s + 2.01e8·26-s + 2.86e8·27-s + 6.88e8·28-s + ⋯
 L(s)  = 1 + 2.82·2-s + 2.30·3-s + 5·4-s + 1.78·5-s + 6.53·6-s + 1.51·7-s + 7.07·8-s + 10/3·9-s + 5.05·10-s + 0.857·11-s + 11.5·12-s + 1.17·13-s + 4.27·14-s + 4.13·15-s + 35/4·16-s + 1.48·17-s + 9.42·18-s + 1.15·19-s + 8.94·20-s + 3.49·21-s + 2.42·22-s + 0.0166·23-s + 16.3·24-s + 2·25-s + 3.32·26-s + 3.84·27-s + 7.55·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$6.77794\times 10^{8}$$ Root analytic conductor: $$12.7024$$ Motivic weight: $$11$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{210} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 11/2, 11/2, 11/2, 11/2 ),\ 1 )$$

## Particular Values

 $$L(6)$$ $$\approx$$ $$1474.690904$$ $$L(\frac12)$$ $$\approx$$ $$1474.690904$$ $$L(\frac{13}{2})$$ 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$C_1$ $$( 1 - p^{5} T )^{4}$$
3$C_1$ $$( 1 - p^{5} T )^{4}$$
5$C_1$ $$( 1 - p^{5} T )^{4}$$
7$C_1$ $$( 1 - p^{5} T )^{4}$$
good11$C_2 \wr S_4$ $$1 - 41660 p T + 555427682412 T^{2} - 6716091718596444 p T^{3} +$$$$13\!\cdots\!22$$$$T^{4} - 6716091718596444 p^{12} T^{5} + 555427682412 p^{22} T^{6} - 41660 p^{34} T^{7} + p^{44} T^{8}$$
13$C_2 \wr S_4$ $$1 - 1574316 T + 4698724538756 T^{2} - 4144806283403141028 T^{3} +$$$$95\!\cdots\!90$$$$T^{4} - 4144806283403141028 p^{11} T^{5} + 4698724538756 p^{22} T^{6} - 1574316 p^{33} T^{7} + p^{44} T^{8}$$
17$C_2 \wr S_4$ $$1 - 8678072 T + 108830083516764 T^{2} -$$$$75\!\cdots\!20$$$$T^{3} +$$$$54\!\cdots\!02$$$$T^{4} -$$$$75\!\cdots\!20$$$$p^{11} T^{5} + 108830083516764 p^{22} T^{6} - 8678072 p^{33} T^{7} + p^{44} T^{8}$$
19$C_2 \wr S_4$ $$1 - 12442004 T + 463934845572076 T^{2} -$$$$42\!\cdots\!28$$$$T^{3} +$$$$80\!\cdots\!66$$$$T^{4} -$$$$42\!\cdots\!28$$$$p^{11} T^{5} + 463934845572076 p^{22} T^{6} - 12442004 p^{33} T^{7} + p^{44} T^{8}$$
23$C_2 \wr S_4$ $$1 - 513088 T + 230211228849468 T^{2} -$$$$24\!\cdots\!04$$$$T^{3} -$$$$48\!\cdots\!86$$$$T^{4} -$$$$24\!\cdots\!04$$$$p^{11} T^{5} + 230211228849468 p^{22} T^{6} - 513088 p^{33} T^{7} + p^{44} T^{8}$$
29$C_2 \wr S_4$ $$1 - 58476696 T + 25009583021169356 T^{2} -$$$$15\!\cdots\!52$$$$T^{3} +$$$$32\!\cdots\!66$$$$T^{4} -$$$$15\!\cdots\!52$$$$p^{11} T^{5} + 25009583021169356 p^{22} T^{6} - 58476696 p^{33} T^{7} + p^{44} T^{8}$$
31$C_2 \wr S_4$ $$1 - 145189572 T + 64640957819813612 T^{2} -$$$$96\!\cdots\!16$$$$T^{3} +$$$$22\!\cdots\!70$$$$T^{4} -$$$$96\!\cdots\!16$$$$p^{11} T^{5} + 64640957819813612 p^{22} T^{6} - 145189572 p^{33} T^{7} + p^{44} T^{8}$$
37$C_2 \wr S_4$ $$1 - 340912752 T + 449679410845919132 T^{2} -$$$$13\!\cdots\!36$$$$T^{3} +$$$$11\!\cdots\!98$$$$T^{4} -$$$$13\!\cdots\!36$$$$p^{11} T^{5} + 449679410845919132 p^{22} T^{6} - 340912752 p^{33} T^{7} + p^{44} T^{8}$$
41$C_2 \wr S_4$ $$1 - 915147368 T + 199627096716234492 T^{2} +$$$$12\!\cdots\!28$$$$T^{3} -$$$$12\!\cdots\!86$$$$T^{4} +$$$$12\!\cdots\!28$$$$p^{11} T^{5} + 199627096716234492 p^{22} T^{6} - 915147368 p^{33} T^{7} + p^{44} T^{8}$$
43$C_2 \wr S_4$ $$1 - 462244024 T - 337499855021694484 T^{2} +$$$$58\!\cdots\!80$$$$T^{3} +$$$$30\!\cdots\!82$$$$T^{4} +$$$$58\!\cdots\!80$$$$p^{11} T^{5} - 337499855021694484 p^{22} T^{6} - 462244024 p^{33} T^{7} + p^{44} T^{8}$$
47$C_2 \wr S_4$ $$1 - 19185320 p T + 3945897582754584156 T^{2} -$$$$70\!\cdots\!32$$$$T^{3} +$$$$11\!\cdots\!10$$$$T^{4} -$$$$70\!\cdots\!32$$$$p^{11} T^{5} + 3945897582754584156 p^{22} T^{6} - 19185320 p^{34} T^{7} + p^{44} T^{8}$$
53$C_2 \wr S_4$ $$1 + 157945788 T + 15178943563572621524 T^{2} -$$$$14\!\cdots\!24$$$$T^{3} +$$$$15\!\cdots\!38$$$$T^{4} -$$$$14\!\cdots\!24$$$$p^{11} T^{5} + 15178943563572621524 p^{22} T^{6} + 157945788 p^{33} T^{7} + p^{44} T^{8}$$
59$C_2 \wr S_4$ $$1 - 2706989128 T +$$$$10\!\cdots\!48$$$$T^{2} -$$$$21\!\cdots\!96$$$$T^{3} +$$$$46\!\cdots\!02$$$$T^{4} -$$$$21\!\cdots\!96$$$$p^{11} T^{5} +$$$$10\!\cdots\!48$$$$p^{22} T^{6} - 2706989128 p^{33} T^{7} + p^{44} T^{8}$$
61$C_2 \wr S_4$ $$1 - 8740846920 T + 63563623150159198700 T^{2} -$$$$31\!\cdots\!76$$$$T^{3} +$$$$30\!\cdots\!10$$$$T^{4} -$$$$31\!\cdots\!76$$$$p^{11} T^{5} + 63563623150159198700 p^{22} T^{6} - 8740846920 p^{33} T^{7} + p^{44} T^{8}$$
67$C_2 \wr S_4$ $$1 - 5883134368 T +$$$$49\!\cdots\!60$$$$T^{2} -$$$$21\!\cdots\!88$$$$T^{3} +$$$$91\!\cdots\!02$$$$T^{4} -$$$$21\!\cdots\!88$$$$p^{11} T^{5} +$$$$49\!\cdots\!60$$$$p^{22} T^{6} - 5883134368 p^{33} T^{7} + p^{44} T^{8}$$
71$C_2 \wr S_4$ $$1 - 344015372 T +$$$$23\!\cdots\!40$$$$T^{2} -$$$$52\!\cdots\!68$$$$T^{3} +$$$$84\!\cdots\!98$$$$T^{4} -$$$$52\!\cdots\!68$$$$p^{11} T^{5} +$$$$23\!\cdots\!40$$$$p^{22} T^{6} - 344015372 p^{33} T^{7} + p^{44} T^{8}$$
73$C_2 \wr S_4$ $$1 - 10549706244 T +$$$$13\!\cdots\!68$$$$T^{2} -$$$$68\!\cdots\!08$$$$T^{3} +$$$$15\!\cdots\!98$$$$T^{4} -$$$$68\!\cdots\!08$$$$p^{11} T^{5} +$$$$13\!\cdots\!68$$$$p^{22} T^{6} - 10549706244 p^{33} T^{7} + p^{44} T^{8}$$
79$C_2 \wr S_4$ $$1 + 430177976 T +$$$$14\!\cdots\!24$$$$T^{2} -$$$$32\!\cdots\!68$$$$T^{3} +$$$$10\!\cdots\!10$$$$T^{4} -$$$$32\!\cdots\!68$$$$p^{11} T^{5} +$$$$14\!\cdots\!24$$$$p^{22} T^{6} + 430177976 p^{33} T^{7} + p^{44} T^{8}$$
83$C_2 \wr S_4$ $$1 - 28504941432 T +$$$$35\!\cdots\!76$$$$T^{2} -$$$$80\!\cdots\!12$$$$T^{3} +$$$$57\!\cdots\!74$$$$T^{4} -$$$$80\!\cdots\!12$$$$p^{11} T^{5} +$$$$35\!\cdots\!76$$$$p^{22} T^{6} - 28504941432 p^{33} T^{7} + p^{44} T^{8}$$
89$C_2 \wr S_4$ $$1 - 26763786680 T +$$$$79\!\cdots\!60$$$$T^{2} -$$$$23\!\cdots\!40$$$$T^{3} +$$$$29\!\cdots\!38$$$$T^{4} -$$$$23\!\cdots\!40$$$$p^{11} T^{5} +$$$$79\!\cdots\!60$$$$p^{22} T^{6} - 26763786680 p^{33} T^{7} + p^{44} T^{8}$$
97$C_2 \wr S_4$ $$1 - 62389990476 T +$$$$25\!\cdots\!84$$$$T^{2} -$$$$13\!\cdots\!04$$$$T^{3} +$$$$25\!\cdots\!34$$$$T^{4} -$$$$13\!\cdots\!04$$$$p^{11} T^{5} +$$$$25\!\cdots\!84$$$$p^{22} T^{6} - 62389990476 p^{33} T^{7} + p^{44} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$