# Properties

 Degree 10 Conductor $1$ Sign $1$ Motivic weight 59 Primitive no Self-dual yes Analytic rank 0

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## Dirichlet series

 L(s)  = 1 − 4.49e8·2-s + 8.40e13·3-s − 4.70e17·4-s + 1.79e20·5-s − 3.77e22·6-s + 1.49e24·7-s + 2.13e26·8-s − 1.54e28·9-s − 8.09e28·10-s + 4.25e30·11-s − 3.95e31·12-s − 8.45e32·13-s − 6.73e32·14-s + 1.51e34·15-s + 4.33e34·16-s − 3.40e36·17-s + 6.95e36·18-s + 6.46e37·19-s − 8.46e37·20-s + 1.25e38·21-s − 1.91e39·22-s + 2.67e40·23-s + 1.79e40·24-s − 1.04e41·25-s + 3.80e41·26-s − 4.87e41·27-s − 7.05e41·28-s + ⋯
 L(s)  = 1 − 0.592·2-s + 0.706·3-s − 0.816·4-s + 0.432·5-s − 0.418·6-s + 0.175·7-s + 0.487·8-s − 1.09·9-s − 0.255·10-s + 0.809·11-s − 0.577·12-s − 1.16·13-s − 0.104·14-s + 0.305·15-s + 0.130·16-s − 1.71·17-s + 0.648·18-s + 1.22·19-s − 0.352·20-s + 0.124·21-s − 0.479·22-s + 1.80·23-s + 0.344·24-s − 0.601·25-s + 0.689·26-s − 0.290·27-s − 0.143·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{5} \, L(s)\cr =\mathstrut & \,\Lambda(60-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+59/2)^{5} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$10$$ $$N$$ = $$1$$ $$\varepsilon$$ = $1$ motivic weight = $$59$$ character : $\chi_{1} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(10,\ 1,\ (\ :59/2, 59/2, 59/2, 59/2, 59/2),\ 1)$ $L(30)$ $\approx$ $0.247569$ $L(\frac12)$ $\approx$ $0.247569$ $L(\frac{61}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, $$F_p$$ is a polynomial of degree 10.
$p$$\Gal(F_p)$$F_p$
good2$C_2 \wr S_5$ $$1 + 56211483 p^{3} T + 328541785870451 p^{11} T^{2} + 35871121944327534405 p^{23} T^{3} +$$$$22\!\cdots\!19$$$$p^{37} T^{4} +$$$$43\!\cdots\!37$$$$p^{56} T^{5} +$$$$22\!\cdots\!19$$$$p^{96} T^{6} + 35871121944327534405 p^{141} T^{7} + 328541785870451 p^{188} T^{8} + 56211483 p^{239} T^{9} + p^{295} T^{10}$$
3$C_2 \wr S_5$ $$1 - 9335181305548 p^{2} T +$$$$10\!\cdots\!01$$$$p^{7} T^{2} -$$$$62\!\cdots\!40$$$$p^{16} T^{3} +$$$$67\!\cdots\!14$$$$p^{27} T^{4} -$$$$31\!\cdots\!96$$$$p^{40} T^{5} +$$$$67\!\cdots\!14$$$$p^{86} T^{6} -$$$$62\!\cdots\!40$$$$p^{134} T^{7} +$$$$10\!\cdots\!01$$$$p^{184} T^{8} - 9335181305548 p^{238} T^{9} + p^{295} T^{10}$$
5$C_2 \wr S_5$ $$1 - 35992322430017211678 p T +$$$$43\!\cdots\!37$$$$p^{5} T^{2} +$$$$34\!\cdots\!52$$$$p^{10} T^{3} +$$$$32\!\cdots\!66$$$$p^{18} T^{4} +$$$$11\!\cdots\!84$$$$p^{27} T^{5} +$$$$32\!\cdots\!66$$$$p^{77} T^{6} +$$$$34\!\cdots\!52$$$$p^{128} T^{7} +$$$$43\!\cdots\!37$$$$p^{182} T^{8} - 35992322430017211678 p^{237} T^{9} + p^{295} T^{10}$$
7$C_2 \wr S_5$ $$1 -$$$$21\!\cdots\!08$$$$p T +$$$$11\!\cdots\!43$$$$p^{4} T^{2} -$$$$10\!\cdots\!00$$$$p^{8} T^{3} +$$$$48\!\cdots\!02$$$$p^{14} T^{4} -$$$$12\!\cdots\!84$$$$p^{21} T^{5} +$$$$48\!\cdots\!02$$$$p^{73} T^{6} -$$$$10\!\cdots\!00$$$$p^{126} T^{7} +$$$$11\!\cdots\!43$$$$p^{181} T^{8} -$$$$21\!\cdots\!08$$$$p^{237} T^{9} + p^{295} T^{10}$$
11$C_2 \wr S_5$ $$1 -$$$$38\!\cdots\!60$$$$p T +$$$$66\!\cdots\!45$$$$p^{3} T^{2} -$$$$20\!\cdots\!20$$$$p^{6} T^{3} +$$$$15\!\cdots\!10$$$$p^{10} T^{4} -$$$$35\!\cdots\!72$$$$p^{14} T^{5} +$$$$15\!\cdots\!10$$$$p^{69} T^{6} -$$$$20\!\cdots\!20$$$$p^{124} T^{7} +$$$$66\!\cdots\!45$$$$p^{180} T^{8} -$$$$38\!\cdots\!60$$$$p^{237} T^{9} + p^{295} T^{10}$$
13$C_2 \wr S_5$ $$1 +$$$$84\!\cdots\!78$$$$T +$$$$13\!\cdots\!09$$$$p T^{2} +$$$$37\!\cdots\!60$$$$p^{3} T^{3} +$$$$25\!\cdots\!82$$$$p^{6} T^{4} +$$$$32\!\cdots\!76$$$$p^{10} T^{5} +$$$$25\!\cdots\!82$$$$p^{65} T^{6} +$$$$37\!\cdots\!60$$$$p^{121} T^{7} +$$$$13\!\cdots\!09$$$$p^{178} T^{8} +$$$$84\!\cdots\!78$$$$p^{236} T^{9} + p^{295} T^{10}$$
17$C_2 \wr S_5$ $$1 +$$$$34\!\cdots\!54$$$$T +$$$$90\!\cdots\!49$$$$p T^{2} +$$$$66\!\cdots\!40$$$$p^{3} T^{3} +$$$$38\!\cdots\!82$$$$p^{6} T^{4} +$$$$12\!\cdots\!76$$$$p^{9} T^{5} +$$$$38\!\cdots\!82$$$$p^{65} T^{6} +$$$$66\!\cdots\!40$$$$p^{121} T^{7} +$$$$90\!\cdots\!49$$$$p^{178} T^{8} +$$$$34\!\cdots\!54$$$$p^{236} T^{9} + p^{295} T^{10}$$
19$C_2 \wr S_5$ $$1 -$$$$34\!\cdots\!00$$$$p T +$$$$36\!\cdots\!95$$$$p^{2} T^{2} -$$$$47\!\cdots\!00$$$$p^{4} T^{3} +$$$$14\!\cdots\!10$$$$p^{6} T^{4} -$$$$14\!\cdots\!00$$$$p^{8} T^{5} +$$$$14\!\cdots\!10$$$$p^{65} T^{6} -$$$$47\!\cdots\!00$$$$p^{122} T^{7} +$$$$36\!\cdots\!95$$$$p^{179} T^{8} -$$$$34\!\cdots\!00$$$$p^{237} T^{9} + p^{295} T^{10}$$
23$C_2 \wr S_5$ $$1 -$$$$26\!\cdots\!12$$$$T +$$$$29\!\cdots\!89$$$$p T^{2} -$$$$11\!\cdots\!60$$$$p^{3} T^{3} +$$$$40\!\cdots\!06$$$$p^{5} T^{4} -$$$$10\!\cdots\!48$$$$p^{7} T^{5} +$$$$40\!\cdots\!06$$$$p^{64} T^{6} -$$$$11\!\cdots\!60$$$$p^{121} T^{7} +$$$$29\!\cdots\!89$$$$p^{178} T^{8} -$$$$26\!\cdots\!12$$$$p^{236} T^{9} + p^{295} T^{10}$$
29$C_2 \wr S_5$ $$1 +$$$$60\!\cdots\!50$$$$p T +$$$$95\!\cdots\!45$$$$p^{2} T^{2} +$$$$39\!\cdots\!00$$$$p^{3} T^{3} +$$$$37\!\cdots\!10$$$$p^{4} T^{4} +$$$$11\!\cdots\!00$$$$p^{5} T^{5} +$$$$37\!\cdots\!10$$$$p^{63} T^{6} +$$$$39\!\cdots\!00$$$$p^{121} T^{7} +$$$$95\!\cdots\!45$$$$p^{179} T^{8} +$$$$60\!\cdots\!50$$$$p^{237} T^{9} + p^{295} T^{10}$$
31$C_2 \wr S_5$ $$1 +$$$$35\!\cdots\!40$$$$T +$$$$91\!\cdots\!95$$$$T^{2} +$$$$50\!\cdots\!80$$$$p T^{3} +$$$$23\!\cdots\!10$$$$p^{2} T^{4} +$$$$80\!\cdots\!28$$$$p^{3} T^{5} +$$$$23\!\cdots\!10$$$$p^{61} T^{6} +$$$$50\!\cdots\!80$$$$p^{119} T^{7} +$$$$91\!\cdots\!95$$$$p^{177} T^{8} +$$$$35\!\cdots\!40$$$$p^{236} T^{9} + p^{295} T^{10}$$
37$C_2 \wr S_5$ $$1 -$$$$12\!\cdots\!26$$$$T +$$$$96\!\cdots\!13$$$$T^{2} -$$$$35\!\cdots\!20$$$$p T^{3} +$$$$36\!\cdots\!62$$$$p^{2} T^{4} -$$$$11\!\cdots\!36$$$$p^{3} T^{5} +$$$$36\!\cdots\!62$$$$p^{61} T^{6} -$$$$35\!\cdots\!20$$$$p^{119} T^{7} +$$$$96\!\cdots\!13$$$$p^{177} T^{8} -$$$$12\!\cdots\!26$$$$p^{236} T^{9} + p^{295} T^{10}$$
41$C_2 \wr S_5$ $$1 +$$$$12\!\cdots\!90$$$$T +$$$$24\!\cdots\!45$$$$p T^{2} +$$$$37\!\cdots\!80$$$$p^{2} T^{3} +$$$$45\!\cdots\!10$$$$p^{3} T^{4} +$$$$44\!\cdots\!68$$$$p^{4} T^{5} +$$$$45\!\cdots\!10$$$$p^{62} T^{6} +$$$$37\!\cdots\!80$$$$p^{120} T^{7} +$$$$24\!\cdots\!45$$$$p^{178} T^{8} +$$$$12\!\cdots\!90$$$$p^{236} T^{9} + p^{295} T^{10}$$
43$C_2 \wr S_5$ $$1 -$$$$13\!\cdots\!92$$$$T +$$$$19\!\cdots\!49$$$$p T^{2} -$$$$41\!\cdots\!00$$$$p^{2} T^{3} +$$$$41\!\cdots\!14$$$$p^{3} T^{4} -$$$$67\!\cdots\!16$$$$p^{4} T^{5} +$$$$41\!\cdots\!14$$$$p^{62} T^{6} -$$$$41\!\cdots\!00$$$$p^{120} T^{7} +$$$$19\!\cdots\!49$$$$p^{178} T^{8} -$$$$13\!\cdots\!92$$$$p^{236} T^{9} + p^{295} T^{10}$$
47$C_2 \wr S_5$ $$1 -$$$$12\!\cdots\!28$$$$p T +$$$$13\!\cdots\!67$$$$p^{2} T^{2} -$$$$89\!\cdots\!40$$$$p^{3} T^{3} +$$$$55\!\cdots\!98$$$$p^{4} T^{4} -$$$$25\!\cdots\!64$$$$p^{5} T^{5} +$$$$55\!\cdots\!98$$$$p^{63} T^{6} -$$$$89\!\cdots\!40$$$$p^{121} T^{7} +$$$$13\!\cdots\!67$$$$p^{179} T^{8} -$$$$12\!\cdots\!28$$$$p^{237} T^{9} + p^{295} T^{10}$$
53$C_2 \wr S_5$ $$1 -$$$$42\!\cdots\!94$$$$p T +$$$$14\!\cdots\!93$$$$p^{2} T^{2} -$$$$32\!\cdots\!20$$$$p^{3} T^{3} +$$$$61\!\cdots\!78$$$$p^{4} T^{4} -$$$$91\!\cdots\!72$$$$p^{5} T^{5} +$$$$61\!\cdots\!78$$$$p^{63} T^{6} -$$$$32\!\cdots\!20$$$$p^{121} T^{7} +$$$$14\!\cdots\!93$$$$p^{179} T^{8} -$$$$42\!\cdots\!94$$$$p^{237} T^{9} + p^{295} T^{10}$$
59$C_2 \wr S_5$ $$1 +$$$$21\!\cdots\!00$$$$T +$$$$79\!\cdots\!95$$$$T^{2} +$$$$13\!\cdots\!00$$$$T^{3} +$$$$32\!\cdots\!10$$$$T^{4} +$$$$40\!\cdots\!00$$$$T^{5} +$$$$32\!\cdots\!10$$$$p^{59} T^{6} +$$$$13\!\cdots\!00$$$$p^{118} T^{7} +$$$$79\!\cdots\!95$$$$p^{177} T^{8} +$$$$21\!\cdots\!00$$$$p^{236} T^{9} + p^{295} T^{10}$$
61$C_2 \wr S_5$ $$1 +$$$$55\!\cdots\!90$$$$T +$$$$84\!\cdots\!45$$$$T^{2} +$$$$33\!\cdots\!80$$$$T^{3} +$$$$29\!\cdots\!10$$$$T^{4} +$$$$92\!\cdots\!48$$$$T^{5} +$$$$29\!\cdots\!10$$$$p^{59} T^{6} +$$$$33\!\cdots\!80$$$$p^{118} T^{7} +$$$$84\!\cdots\!45$$$$p^{177} T^{8} +$$$$55\!\cdots\!90$$$$p^{236} T^{9} + p^{295} T^{10}$$
67$C_2 \wr S_5$ $$1 -$$$$25\!\cdots\!96$$$$T +$$$$20\!\cdots\!83$$$$T^{2} -$$$$32\!\cdots\!80$$$$T^{3} +$$$$19\!\cdots\!58$$$$T^{4} -$$$$22\!\cdots\!28$$$$T^{5} +$$$$19\!\cdots\!58$$$$p^{59} T^{6} -$$$$32\!\cdots\!80$$$$p^{118} T^{7} +$$$$20\!\cdots\!83$$$$p^{177} T^{8} -$$$$25\!\cdots\!96$$$$p^{236} T^{9} + p^{295} T^{10}$$
71$C_2 \wr S_5$ $$1 +$$$$63\!\cdots\!40$$$$T +$$$$82\!\cdots\!95$$$$T^{2} +$$$$37\!\cdots\!80$$$$T^{3} +$$$$26\!\cdots\!10$$$$T^{4} +$$$$89\!\cdots\!48$$$$T^{5} +$$$$26\!\cdots\!10$$$$p^{59} T^{6} +$$$$37\!\cdots\!80$$$$p^{118} T^{7} +$$$$82\!\cdots\!95$$$$p^{177} T^{8} +$$$$63\!\cdots\!40$$$$p^{236} T^{9} + p^{295} T^{10}$$
73$C_2 \wr S_5$ $$1 -$$$$12\!\cdots\!62$$$$T +$$$$31\!\cdots\!97$$$$T^{2} -$$$$37\!\cdots\!20$$$$T^{3} +$$$$48\!\cdots\!58$$$$T^{4} -$$$$45\!\cdots\!56$$$$T^{5} +$$$$48\!\cdots\!58$$$$p^{59} T^{6} -$$$$37\!\cdots\!20$$$$p^{118} T^{7} +$$$$31\!\cdots\!97$$$$p^{177} T^{8} -$$$$12\!\cdots\!62$$$$p^{236} T^{9} + p^{295} T^{10}$$
79$C_2 \wr S_5$ $$1 +$$$$19\!\cdots\!00$$$$T +$$$$33\!\cdots\!95$$$$T^{2} +$$$$34\!\cdots\!00$$$$T^{3} +$$$$36\!\cdots\!10$$$$T^{4} +$$$$30\!\cdots\!00$$$$T^{5} +$$$$36\!\cdots\!10$$$$p^{59} T^{6} +$$$$34\!\cdots\!00$$$$p^{118} T^{7} +$$$$33\!\cdots\!95$$$$p^{177} T^{8} +$$$$19\!\cdots\!00$$$$p^{236} T^{9} + p^{295} T^{10}$$
83$C_2 \wr S_5$ $$1 -$$$$13\!\cdots\!52$$$$T +$$$$14\!\cdots\!27$$$$T^{2} -$$$$10\!\cdots\!60$$$$T^{3} +$$$$59\!\cdots\!78$$$$T^{4} -$$$$26\!\cdots\!36$$$$T^{5} +$$$$59\!\cdots\!78$$$$p^{59} T^{6} -$$$$10\!\cdots\!60$$$$p^{118} T^{7} +$$$$14\!\cdots\!27$$$$p^{177} T^{8} -$$$$13\!\cdots\!52$$$$p^{236} T^{9} + p^{295} T^{10}$$
89$C_2 \wr S_5$ $$1 -$$$$41\!\cdots\!50$$$$T +$$$$38\!\cdots\!45$$$$T^{2} -$$$$10\!\cdots\!00$$$$T^{3} +$$$$63\!\cdots\!10$$$$T^{4} -$$$$13\!\cdots\!00$$$$T^{5} +$$$$63\!\cdots\!10$$$$p^{59} T^{6} -$$$$10\!\cdots\!00$$$$p^{118} T^{7} +$$$$38\!\cdots\!45$$$$p^{177} T^{8} -$$$$41\!\cdots\!50$$$$p^{236} T^{9} + p^{295} T^{10}$$
97$C_2 \wr S_5$ $$1 -$$$$11\!\cdots\!66$$$$T +$$$$92\!\cdots\!53$$$$T^{2} -$$$$60\!\cdots\!20$$$$T^{3} +$$$$32\!\cdots\!38$$$$T^{4} -$$$$13\!\cdots\!48$$$$T^{5} +$$$$32\!\cdots\!38$$$$p^{59} T^{6} -$$$$60\!\cdots\!20$$$$p^{118} T^{7} +$$$$92\!\cdots\!53$$$$p^{177} T^{8} -$$$$11\!\cdots\!66$$$$p^{236} T^{9} + p^{295} T^{10}$$
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\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

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

−9.423065804145231908188850616699, −9.245452262231743604694207948846, −9.060582200406625685943185371625, −8.802134135045037918398201960027, −8.698606407186667715640325704438, −7.80551940878668052863160991095, −7.53689042800403740235672257470, −7.14358773262112345305516870612, −6.93945325990773163531099102392, −6.19121117951191906941621640811, −5.85938177333788607586037303447, −5.32663091084914999311716507641, −5.03166302061618668622875081168, −4.82438506927423381864202585868, −4.26781622403418384419977402297, −3.55585079515958630395106621156, −3.42616382507351596843680429771, −3.25519075822932993127530371829, −2.43010475932392204575042668200, −2.11862684631809459824583932265, −2.02539191838660150640447250153, −1.39111825076257501596238050121, −1.03375234070354278860348613847, −0.38173724720063210159473700199, −0.11548805786913623004689854185, 0.11548805786913623004689854185, 0.38173724720063210159473700199, 1.03375234070354278860348613847, 1.39111825076257501596238050121, 2.02539191838660150640447250153, 2.11862684631809459824583932265, 2.43010475932392204575042668200, 3.25519075822932993127530371829, 3.42616382507351596843680429771, 3.55585079515958630395106621156, 4.26781622403418384419977402297, 4.82438506927423381864202585868, 5.03166302061618668622875081168, 5.32663091084914999311716507641, 5.85938177333788607586037303447, 6.19121117951191906941621640811, 6.93945325990773163531099102392, 7.14358773262112345305516870612, 7.53689042800403740235672257470, 7.80551940878668052863160991095, 8.698606407186667715640325704438, 8.802134135045037918398201960027, 9.060582200406625685943185371625, 9.245452262231743604694207948846, 9.423065804145231908188850616699