# Properties

 Label 10-1-1.1-c63e5-0-0 Degree $10$ Conductor $1$ Sign $1$ Analytic cond. $1.00343\times 10^{7}$ Root an. cond. $5.01359$ Motivic weight $63$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 5.07e8·2-s + 9.53e14·3-s − 1.95e19·4-s − 5.01e20·5-s + 4.83e23·6-s + 3.76e26·7-s − 9.06e27·8-s − 2.08e30·9-s − 2.54e29·10-s − 5.40e32·11-s − 1.86e34·12-s + 1.08e35·13-s + 1.91e35·14-s − 4.77e35·15-s + 1.72e38·16-s + 2.33e38·17-s − 1.05e39·18-s − 7.86e39·19-s + 9.79e39·20-s + 3.59e41·21-s − 2.74e41·22-s + 1.57e43·23-s − 8.63e42·24-s − 1.24e44·25-s + 5.48e43·26-s − 6.44e44·27-s − 7.36e45·28-s + ⋯
 L(s)  = 1 + 0.167·2-s + 0.891·3-s − 2.11·4-s − 0.0481·5-s + 0.148·6-s + 0.902·7-s − 0.323·8-s − 1.82·9-s − 0.00804·10-s − 0.848·11-s − 1.88·12-s + 0.880·13-s + 0.150·14-s − 0.0428·15-s + 2.02·16-s + 0.406·17-s − 0.304·18-s − 0.412·19-s + 0.101·20-s + 0.804·21-s − 0.141·22-s + 2.00·23-s − 0.288·24-s − 1.15·25-s + 0.147·26-s − 0.526·27-s − 1.91·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(64-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+63/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$10$$ Conductor: $$1$$ Sign: $1$ Analytic conductor: $$1.00343\times 10^{7}$$ Root analytic conductor: $$5.01359$$ Motivic weight: $$63$$ Rational: yes Arithmetic: yes Character: $\chi_{1} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(10,\ 1,\ (\ :63/2, 63/2, 63/2, 63/2, 63/2),\ 1)$$

## Particular Values

 $$L(32)$$ $$\approx$$ $$8.712291760$$ $$L(\frac12)$$ $$\approx$$ $$8.712291760$$ $$L(\frac{65}{2})$$ 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)$
good2$C_2 \wr S_5$ $$1 - 63414387 p^{3} T + 19336575150992617 p^{10} T^{2} -$$$$12\!\cdots\!25$$$$p^{23} T^{3} +$$$$78\!\cdots\!87$$$$p^{38} T^{4} -$$$$15\!\cdots\!91$$$$p^{55} T^{5} +$$$$78\!\cdots\!87$$$$p^{101} T^{6} -$$$$12\!\cdots\!25$$$$p^{149} T^{7} + 19336575150992617 p^{199} T^{8} - 63414387 p^{255} T^{9} + p^{315} T^{10}$$
3$C_2 \wr S_5$ $$1 - 3922820374964 p^{5} T +$$$$45\!\cdots\!07$$$$p^{8} T^{2} -$$$$32\!\cdots\!00$$$$p^{17} T^{3} +$$$$24\!\cdots\!42$$$$p^{30} T^{4} -$$$$20\!\cdots\!08$$$$p^{43} T^{5} +$$$$24\!\cdots\!42$$$$p^{93} T^{6} -$$$$32\!\cdots\!00$$$$p^{143} T^{7} +$$$$45\!\cdots\!07$$$$p^{197} T^{8} - 3922820374964 p^{257} T^{9} + p^{315} T^{10}$$
5$C_2 \wr S_5$ $$1 +$$$$10\!\cdots\!86$$$$p T +$$$$20\!\cdots\!49$$$$p^{4} T^{2} -$$$$14\!\cdots\!72$$$$p^{9} T^{3} +$$$$27\!\cdots\!94$$$$p^{17} T^{4} +$$$$75\!\cdots\!76$$$$p^{27} T^{5} +$$$$27\!\cdots\!94$$$$p^{80} T^{6} -$$$$14\!\cdots\!72$$$$p^{135} T^{7} +$$$$20\!\cdots\!49$$$$p^{193} T^{8} +$$$$10\!\cdots\!86$$$$p^{253} T^{9} + p^{315} T^{10}$$
7$C_2 \wr S_5$ $$1 -$$$$53\!\cdots\!08$$$$p T +$$$$20\!\cdots\!49$$$$p^{5} T^{2} -$$$$11\!\cdots\!00$$$$p^{10} T^{3} +$$$$75\!\cdots\!98$$$$p^{16} T^{4} +$$$$30\!\cdots\!84$$$$p^{23} T^{5} +$$$$75\!\cdots\!98$$$$p^{79} T^{6} -$$$$11\!\cdots\!00$$$$p^{136} T^{7} +$$$$20\!\cdots\!49$$$$p^{194} T^{8} -$$$$53\!\cdots\!08$$$$p^{253} T^{9} + p^{315} T^{10}$$
11$C_2 \wr S_5$ $$1 +$$$$49\!\cdots\!40$$$$p T +$$$$10\!\cdots\!95$$$$p^{2} T^{2} +$$$$26\!\cdots\!80$$$$p^{5} T^{3} +$$$$29\!\cdots\!10$$$$p^{9} T^{4} +$$$$47\!\cdots\!28$$$$p^{14} T^{5} +$$$$29\!\cdots\!10$$$$p^{72} T^{6} +$$$$26\!\cdots\!80$$$$p^{131} T^{7} +$$$$10\!\cdots\!95$$$$p^{191} T^{8} +$$$$49\!\cdots\!40$$$$p^{253} T^{9} + p^{315} T^{10}$$
13$C_2 \wr S_5$ $$1 -$$$$83\!\cdots\!74$$$$p T +$$$$15\!\cdots\!13$$$$p^{2} T^{2} -$$$$99\!\cdots\!00$$$$p^{3} T^{3} +$$$$12\!\cdots\!02$$$$p^{6} T^{4} -$$$$45\!\cdots\!84$$$$p^{10} T^{5} +$$$$12\!\cdots\!02$$$$p^{69} T^{6} -$$$$99\!\cdots\!00$$$$p^{129} T^{7} +$$$$15\!\cdots\!13$$$$p^{191} T^{8} -$$$$83\!\cdots\!74$$$$p^{253} T^{9} + p^{315} T^{10}$$
17$C_2 \wr S_5$ $$1 -$$$$23\!\cdots\!26$$$$T +$$$$48\!\cdots\!89$$$$p T^{2} -$$$$59\!\cdots\!00$$$$p^{3} T^{3} +$$$$32\!\cdots\!34$$$$p^{5} T^{4} -$$$$15\!\cdots\!68$$$$p^{8} T^{5} +$$$$32\!\cdots\!34$$$$p^{68} T^{6} -$$$$59\!\cdots\!00$$$$p^{129} T^{7} +$$$$48\!\cdots\!89$$$$p^{190} T^{8} -$$$$23\!\cdots\!26$$$$p^{252} T^{9} + p^{315} T^{10}$$
19$C_2 \wr S_5$ $$1 +$$$$41\!\cdots\!00$$$$p T +$$$$40\!\cdots\!95$$$$p^{2} T^{2} +$$$$86\!\cdots\!00$$$$p^{4} T^{3} +$$$$10\!\cdots\!90$$$$p^{7} T^{4} +$$$$10\!\cdots\!00$$$$p^{10} T^{5} +$$$$10\!\cdots\!90$$$$p^{70} T^{6} +$$$$86\!\cdots\!00$$$$p^{130} T^{7} +$$$$40\!\cdots\!95$$$$p^{191} T^{8} +$$$$41\!\cdots\!00$$$$p^{253} T^{9} + p^{315} T^{10}$$
23$C_2 \wr S_5$ $$1 -$$$$15\!\cdots\!72$$$$T +$$$$11\!\cdots\!29$$$$p T^{2} -$$$$36\!\cdots\!00$$$$p^{2} T^{3} +$$$$66\!\cdots\!58$$$$p^{4} T^{4} -$$$$66\!\cdots\!44$$$$p^{6} T^{5} +$$$$66\!\cdots\!58$$$$p^{67} T^{6} -$$$$36\!\cdots\!00$$$$p^{128} T^{7} +$$$$11\!\cdots\!29$$$$p^{190} T^{8} -$$$$15\!\cdots\!72$$$$p^{252} T^{9} + p^{315} T^{10}$$
29$C_2 \wr S_5$ $$1 -$$$$17\!\cdots\!50$$$$p T +$$$$19\!\cdots\!45$$$$p^{2} T^{2} -$$$$14\!\cdots\!00$$$$p^{3} T^{3} +$$$$29\!\cdots\!90$$$$p^{5} T^{4} -$$$$45\!\cdots\!00$$$$p^{7} T^{5} +$$$$29\!\cdots\!90$$$$p^{68} T^{6} -$$$$14\!\cdots\!00$$$$p^{129} T^{7} +$$$$19\!\cdots\!45$$$$p^{191} T^{8} -$$$$17\!\cdots\!50$$$$p^{253} T^{9} + p^{315} T^{10}$$
31$C_2 \wr S_5$ $$1 -$$$$16\!\cdots\!60$$$$p^{2} T +$$$$38\!\cdots\!95$$$$p^{2} T^{2} -$$$$13\!\cdots\!20$$$$p^{3} T^{3} +$$$$65\!\cdots\!10$$$$p^{4} T^{4} -$$$$17\!\cdots\!52$$$$p^{5} T^{5} +$$$$65\!\cdots\!10$$$$p^{67} T^{6} -$$$$13\!\cdots\!20$$$$p^{129} T^{7} +$$$$38\!\cdots\!95$$$$p^{191} T^{8} -$$$$16\!\cdots\!60$$$$p^{254} T^{9} + p^{315} T^{10}$$
37$C_2 \wr S_5$ $$1 +$$$$17\!\cdots\!34$$$$T +$$$$27\!\cdots\!53$$$$T^{2} +$$$$10\!\cdots\!00$$$$p T^{3} +$$$$23\!\cdots\!22$$$$p^{2} T^{4} +$$$$67\!\cdots\!04$$$$p^{3} T^{5} +$$$$23\!\cdots\!22$$$$p^{65} T^{6} +$$$$10\!\cdots\!00$$$$p^{127} T^{7} +$$$$27\!\cdots\!53$$$$p^{189} T^{8} +$$$$17\!\cdots\!34$$$$p^{252} T^{9} + p^{315} T^{10}$$
41$C_2 \wr S_5$ $$1 -$$$$15\!\cdots\!10$$$$T +$$$$26\!\cdots\!45$$$$T^{2} -$$$$61\!\cdots\!20$$$$p T^{3} +$$$$14\!\cdots\!10$$$$p^{2} T^{4} -$$$$21\!\cdots\!12$$$$p^{3} T^{5} +$$$$14\!\cdots\!10$$$$p^{65} T^{6} -$$$$61\!\cdots\!20$$$$p^{127} T^{7} +$$$$26\!\cdots\!45$$$$p^{189} T^{8} -$$$$15\!\cdots\!10$$$$p^{252} T^{9} + p^{315} T^{10}$$
43$C_2 \wr S_5$ $$1 +$$$$29\!\cdots\!08$$$$T +$$$$83\!\cdots\!49$$$$p T^{2} +$$$$47\!\cdots\!00$$$$p^{2} T^{3} +$$$$70\!\cdots\!14$$$$p^{3} T^{4} +$$$$30\!\cdots\!84$$$$p^{4} T^{5} +$$$$70\!\cdots\!14$$$$p^{66} T^{6} +$$$$47\!\cdots\!00$$$$p^{128} T^{7} +$$$$83\!\cdots\!49$$$$p^{190} T^{8} +$$$$29\!\cdots\!08$$$$p^{252} T^{9} + p^{315} T^{10}$$
47$C_2 \wr S_5$ $$1 +$$$$48\!\cdots\!64$$$$T +$$$$23\!\cdots\!09$$$$p T^{2} +$$$$18\!\cdots\!00$$$$p^{2} T^{3} +$$$$47\!\cdots\!46$$$$p^{3} T^{4} +$$$$27\!\cdots\!52$$$$p^{4} T^{5} +$$$$47\!\cdots\!46$$$$p^{66} T^{6} +$$$$18\!\cdots\!00$$$$p^{128} T^{7} +$$$$23\!\cdots\!09$$$$p^{190} T^{8} +$$$$48\!\cdots\!64$$$$p^{252} T^{9} + p^{315} T^{10}$$
53$C_2 \wr S_5$ $$1 +$$$$69\!\cdots\!98$$$$T +$$$$24\!\cdots\!09$$$$p T^{2} +$$$$88\!\cdots\!00$$$$p^{2} T^{3} +$$$$48\!\cdots\!54$$$$p^{3} T^{4} -$$$$49\!\cdots\!36$$$$p^{4} T^{5} +$$$$48\!\cdots\!54$$$$p^{66} T^{6} +$$$$88\!\cdots\!00$$$$p^{128} T^{7} +$$$$24\!\cdots\!09$$$$p^{190} T^{8} +$$$$69\!\cdots\!98$$$$p^{252} T^{9} + p^{315} T^{10}$$
59$C_2 \wr S_5$ $$1 -$$$$17\!\cdots\!00$$$$p T +$$$$51\!\cdots\!95$$$$p^{2} T^{2} -$$$$63\!\cdots\!00$$$$p^{3} T^{3} +$$$$10\!\cdots\!10$$$$p^{4} T^{4} -$$$$94\!\cdots\!00$$$$p^{5} T^{5} +$$$$10\!\cdots\!10$$$$p^{67} T^{6} -$$$$63\!\cdots\!00$$$$p^{129} T^{7} +$$$$51\!\cdots\!95$$$$p^{191} T^{8} -$$$$17\!\cdots\!00$$$$p^{253} T^{9} + p^{315} T^{10}$$
61$C_2 \wr S_5$ $$1 -$$$$64\!\cdots\!10$$$$p T +$$$$26\!\cdots\!45$$$$p^{2} T^{2} -$$$$57\!\cdots\!20$$$$p^{3} T^{3} +$$$$12\!\cdots\!10$$$$p^{4} T^{4} -$$$$23\!\cdots\!52$$$$p^{5} T^{5} +$$$$12\!\cdots\!10$$$$p^{67} T^{6} -$$$$57\!\cdots\!20$$$$p^{129} T^{7} +$$$$26\!\cdots\!45$$$$p^{191} T^{8} -$$$$64\!\cdots\!10$$$$p^{253} T^{9} + p^{315} T^{10}$$
67$C_2 \wr S_5$ $$1 +$$$$47\!\cdots\!24$$$$T +$$$$56\!\cdots\!63$$$$T^{2} +$$$$19\!\cdots\!00$$$$T^{3} +$$$$12\!\cdots\!38$$$$T^{4} +$$$$31\!\cdots\!12$$$$T^{5} +$$$$12\!\cdots\!38$$$$p^{63} T^{6} +$$$$19\!\cdots\!00$$$$p^{126} T^{7} +$$$$56\!\cdots\!63$$$$p^{189} T^{8} +$$$$47\!\cdots\!24$$$$p^{252} T^{9} + p^{315} T^{10}$$
71$C_2 \wr S_5$ $$1 -$$$$50\!\cdots\!60$$$$T +$$$$23\!\cdots\!95$$$$T^{2} -$$$$62\!\cdots\!20$$$$T^{3} +$$$$16\!\cdots\!10$$$$T^{4} -$$$$32\!\cdots\!52$$$$T^{5} +$$$$16\!\cdots\!10$$$$p^{63} T^{6} -$$$$62\!\cdots\!20$$$$p^{126} T^{7} +$$$$23\!\cdots\!95$$$$p^{189} T^{8} -$$$$50\!\cdots\!60$$$$p^{252} T^{9} + p^{315} T^{10}$$
73$C_2 \wr S_5$ $$1 +$$$$29\!\cdots\!78$$$$T +$$$$69\!\cdots\!17$$$$T^{2} +$$$$36\!\cdots\!00$$$$T^{3} +$$$$23\!\cdots\!78$$$$T^{4} +$$$$13\!\cdots\!84$$$$T^{5} +$$$$23\!\cdots\!78$$$$p^{63} T^{6} +$$$$36\!\cdots\!00$$$$p^{126} T^{7} +$$$$69\!\cdots\!17$$$$p^{189} T^{8} +$$$$29\!\cdots\!78$$$$p^{252} T^{9} + p^{315} T^{10}$$
79$C_2 \wr S_5$ $$1 -$$$$58\!\cdots\!00$$$$T +$$$$13\!\cdots\!95$$$$T^{2} -$$$$81\!\cdots\!00$$$$T^{3} +$$$$81\!\cdots\!10$$$$T^{4} -$$$$43\!\cdots\!00$$$$T^{5} +$$$$81\!\cdots\!10$$$$p^{63} T^{6} -$$$$81\!\cdots\!00$$$$p^{126} T^{7} +$$$$13\!\cdots\!95$$$$p^{189} T^{8} -$$$$58\!\cdots\!00$$$$p^{252} T^{9} + p^{315} T^{10}$$
83$C_2 \wr S_5$ $$1 -$$$$27\!\cdots\!32$$$$T +$$$$25\!\cdots\!87$$$$T^{2} -$$$$58\!\cdots\!00$$$$T^{3} +$$$$33\!\cdots\!38$$$$T^{4} -$$$$58\!\cdots\!16$$$$T^{5} +$$$$33\!\cdots\!38$$$$p^{63} T^{6} -$$$$58\!\cdots\!00$$$$p^{126} T^{7} +$$$$25\!\cdots\!87$$$$p^{189} T^{8} -$$$$27\!\cdots\!32$$$$p^{252} T^{9} + p^{315} T^{10}$$
89$C_2 \wr S_5$ $$1 -$$$$32\!\cdots\!50$$$$T +$$$$78\!\cdots\!45$$$$T^{2} -$$$$14\!\cdots\!00$$$$T^{3} +$$$$71\!\cdots\!10$$$$T^{4} -$$$$11\!\cdots\!00$$$$T^{5} +$$$$71\!\cdots\!10$$$$p^{63} T^{6} -$$$$14\!\cdots\!00$$$$p^{126} T^{7} +$$$$78\!\cdots\!45$$$$p^{189} T^{8} -$$$$32\!\cdots\!50$$$$p^{252} T^{9} + p^{315} T^{10}$$
97$C_2 \wr S_5$ $$1 +$$$$17\!\cdots\!14$$$$T +$$$$52\!\cdots\!73$$$$T^{2} +$$$$12\!\cdots\!00$$$$T^{3} +$$$$12\!\cdots\!58$$$$T^{4} +$$$$29\!\cdots\!12$$$$T^{5} +$$$$12\!\cdots\!58$$$$p^{63} T^{6} +$$$$12\!\cdots\!00$$$$p^{126} T^{7} +$$$$52\!\cdots\!73$$$$p^{189} T^{8} +$$$$17\!\cdots\!14$$$$p^{252} T^{9} + p^{315} T^{10}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−9.267756777022281022914557827508, −8.572137877718424829530249086383, −8.529931552949539470805252488505, −8.528091325463947058665729713447, −8.521125339432963339889268140313, −7.71513151385096268331106059192, −7.54164017761432726094566132719, −6.47207300135663117431492117326, −6.36932168417662856012186571069, −6.11308787721379987450502308199, −5.19326363783780132194148380768, −5.07839043795559135938970624058, −4.89862718473805193834153393419, −4.64415932938404842791853805292, −4.18209228791014334376063149687, −3.65374290931023038947315699063, −3.13322097782405085288527911067, −3.01726640481319240814008038374, −2.80685379906930605441069033119, −2.21468931928119782116171883813, −1.90718296516908783667460242487, −1.07806871839496369775651965234, −0.64534129392468597418821265851, −0.62694026687984783057215458063, −0.60824199369184346752501954201, 0.60824199369184346752501954201, 0.62694026687984783057215458063, 0.64534129392468597418821265851, 1.07806871839496369775651965234, 1.90718296516908783667460242487, 2.21468931928119782116171883813, 2.80685379906930605441069033119, 3.01726640481319240814008038374, 3.13322097782405085288527911067, 3.65374290931023038947315699063, 4.18209228791014334376063149687, 4.64415932938404842791853805292, 4.89862718473805193834153393419, 5.07839043795559135938970624058, 5.19326363783780132194148380768, 6.11308787721379987450502308199, 6.36932168417662856012186571069, 6.47207300135663117431492117326, 7.54164017761432726094566132719, 7.71513151385096268331106059192, 8.521125339432963339889268140313, 8.528091325463947058665729713447, 8.529931552949539470805252488505, 8.572137877718424829530249086383, 9.267756777022281022914557827508