# Properties

 Degree 14 Conductor $1$ Sign $-1$ Motivic weight 89 Primitive no Self-dual yes Analytic rank 7

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3.14e13·2-s − 1.35e21·3-s − 5.42e26·4-s + 1.02e31·5-s + 4.27e34·6-s + 3.85e37·7-s + 2.78e40·8-s − 6.43e42·9-s − 3.21e44·10-s − 3.31e46·11-s + 7.37e47·12-s − 1.03e50·13-s − 1.20e51·14-s − 1.39e52·15-s − 8.86e51·16-s + 8.31e54·17-s + 2.02e56·18-s + 5.66e56·19-s − 5.54e57·20-s − 5.23e58·21-s + 1.04e60·22-s − 1.16e60·23-s − 3.78e61·24-s − 6.19e62·25-s + 3.24e63·26-s + 1.46e64·27-s − 2.08e64·28-s + ⋯
 L(s)  = 1 − 1.26·2-s − 0.797·3-s − 0.876·4-s + 0.805·5-s + 1.00·6-s + 0.951·7-s + 1.80·8-s − 2.21·9-s − 1.01·10-s − 1.50·11-s + 0.698·12-s − 2.77·13-s − 1.20·14-s − 0.641·15-s − 0.0231·16-s + 1.46·17-s + 2.79·18-s + 0.705·19-s − 0.705·20-s − 0.758·21-s + 1.90·22-s − 0.294·23-s − 1.44·24-s − 3.83·25-s + 3.50·26-s + 2.95·27-s − 0.834·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(90-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+89/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$14$$ $$N$$ = $$1$$ $$\varepsilon$$ = $-1$ motivic weight = $$89$$ character : $\chi_{1} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$7$$ Selberg data = $$(14,\ 1,\ (\ :[89/2]^{7}),\ -1)$$ $$L(45)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{91}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where,$$F_p(T)$$ is a polynomial of degree 14.
$p$$F_p(T)$
good2 $$1 + 1962958146963 p^{4} T +$$$$46\!\cdots\!29$$$$p^{15} T^{2} +$$$$27\!\cdots\!35$$$$p^{27} T^{3} +$$$$10\!\cdots\!27$$$$p^{40} T^{4} +$$$$10\!\cdots\!03$$$$p^{61} T^{5} +$$$$63\!\cdots\!51$$$$p^{86} T^{6} +$$$$57\!\cdots\!35$$$$p^{114} T^{7} +$$$$63\!\cdots\!51$$$$p^{175} T^{8} +$$$$10\!\cdots\!03$$$$p^{239} T^{9} +$$$$10\!\cdots\!27$$$$p^{307} T^{10} +$$$$27\!\cdots\!35$$$$p^{383} T^{11} +$$$$46\!\cdots\!29$$$$p^{460} T^{12} + 1962958146963 p^{538} T^{13} + p^{623} T^{14}$$
3 $$1 +$$$$15\!\cdots\!96$$$$p^{2} T +$$$$12\!\cdots\!73$$$$p^{8} T^{2} +$$$$12\!\cdots\!80$$$$p^{16} T^{3} +$$$$25\!\cdots\!71$$$$p^{27} T^{4} -$$$$12\!\cdots\!56$$$$p^{43} T^{5} +$$$$15\!\cdots\!21$$$$p^{60} T^{6} -$$$$27\!\cdots\!40$$$$p^{78} T^{7} +$$$$15\!\cdots\!21$$$$p^{149} T^{8} -$$$$12\!\cdots\!56$$$$p^{221} T^{9} +$$$$25\!\cdots\!71$$$$p^{294} T^{10} +$$$$12\!\cdots\!80$$$$p^{372} T^{11} +$$$$12\!\cdots\!73$$$$p^{453} T^{12} +$$$$15\!\cdots\!96$$$$p^{536} T^{13} + p^{623} T^{14}$$
5 $$1 -$$$$81\!\cdots\!58$$$$p^{3} T +$$$$92\!\cdots\!43$$$$p^{7} T^{2} -$$$$27\!\cdots\!88$$$$p^{12} T^{3} +$$$$50\!\cdots\!09$$$$p^{21} T^{4} -$$$$43\!\cdots\!22$$$$p^{31} T^{5} +$$$$89\!\cdots\!39$$$$p^{44} T^{6} -$$$$11\!\cdots\!64$$$$p^{58} T^{7} +$$$$89\!\cdots\!39$$$$p^{133} T^{8} -$$$$43\!\cdots\!22$$$$p^{209} T^{9} +$$$$50\!\cdots\!09$$$$p^{288} T^{10} -$$$$27\!\cdots\!88$$$$p^{368} T^{11} +$$$$92\!\cdots\!43$$$$p^{452} T^{12} -$$$$81\!\cdots\!58$$$$p^{537} T^{13} + p^{623} T^{14}$$
7 $$1 -$$$$78\!\cdots\!08$$$$p^{2} T +$$$$33\!\cdots\!57$$$$p^{4} T^{2} -$$$$23\!\cdots\!00$$$$p^{6} T^{3} +$$$$15\!\cdots\!79$$$$p^{11} T^{4} -$$$$57\!\cdots\!76$$$$p^{18} T^{5} +$$$$81\!\cdots\!21$$$$p^{26} T^{6} -$$$$72\!\cdots\!00$$$$p^{36} T^{7} +$$$$81\!\cdots\!21$$$$p^{115} T^{8} -$$$$57\!\cdots\!76$$$$p^{196} T^{9} +$$$$15\!\cdots\!79$$$$p^{278} T^{10} -$$$$23\!\cdots\!00$$$$p^{362} T^{11} +$$$$33\!\cdots\!57$$$$p^{449} T^{12} -$$$$78\!\cdots\!08$$$$p^{536} T^{13} + p^{623} T^{14}$$
11 $$1 +$$$$30\!\cdots\!96$$$$p T +$$$$13\!\cdots\!51$$$$p^{3} T^{2} +$$$$28\!\cdots\!36$$$$p^{6} T^{3} +$$$$74\!\cdots\!81$$$$p^{10} T^{4} +$$$$11\!\cdots\!28$$$$p^{14} T^{5} +$$$$21\!\cdots\!43$$$$p^{19} T^{6} +$$$$23\!\cdots\!88$$$$p^{25} T^{7} +$$$$21\!\cdots\!43$$$$p^{108} T^{8} +$$$$11\!\cdots\!28$$$$p^{192} T^{9} +$$$$74\!\cdots\!81$$$$p^{277} T^{10} +$$$$28\!\cdots\!36$$$$p^{362} T^{11} +$$$$13\!\cdots\!51$$$$p^{448} T^{12} +$$$$30\!\cdots\!96$$$$p^{535} T^{13} + p^{623} T^{14}$$
13 $$1 +$$$$79\!\cdots\!18$$$$p T +$$$$36\!\cdots\!79$$$$p^{3} T^{2} +$$$$98\!\cdots\!20$$$$p^{5} T^{3} +$$$$20\!\cdots\!21$$$$p^{7} T^{4} +$$$$17\!\cdots\!42$$$$p^{10} T^{5} +$$$$27\!\cdots\!03$$$$p^{15} T^{6} -$$$$53\!\cdots\!40$$$$p^{21} T^{7} +$$$$27\!\cdots\!03$$$$p^{104} T^{8} +$$$$17\!\cdots\!42$$$$p^{188} T^{9} +$$$$20\!\cdots\!21$$$$p^{274} T^{10} +$$$$98\!\cdots\!20$$$$p^{361} T^{11} +$$$$36\!\cdots\!79$$$$p^{448} T^{12} +$$$$79\!\cdots\!18$$$$p^{535} T^{13} + p^{623} T^{14}$$
17 $$1 -$$$$48\!\cdots\!26$$$$p T +$$$$32\!\cdots\!79$$$$p^{3} T^{2} -$$$$86\!\cdots\!80$$$$p^{5} T^{3} +$$$$17\!\cdots\!57$$$$p^{8} T^{4} -$$$$14\!\cdots\!94$$$$p^{12} T^{5} +$$$$12\!\cdots\!39$$$$p^{16} T^{6} -$$$$83\!\cdots\!80$$$$p^{20} T^{7} +$$$$12\!\cdots\!39$$$$p^{105} T^{8} -$$$$14\!\cdots\!94$$$$p^{190} T^{9} +$$$$17\!\cdots\!57$$$$p^{275} T^{10} -$$$$86\!\cdots\!80$$$$p^{361} T^{11} +$$$$32\!\cdots\!79$$$$p^{448} T^{12} -$$$$48\!\cdots\!26$$$$p^{535} T^{13} + p^{623} T^{14}$$
19 $$1 -$$$$56\!\cdots\!80$$$$T +$$$$14\!\cdots\!87$$$$p T^{2} -$$$$44\!\cdots\!20$$$$p^{2} T^{3} +$$$$30\!\cdots\!41$$$$p^{4} T^{4} -$$$$23\!\cdots\!00$$$$p^{7} T^{5} +$$$$61\!\cdots\!65$$$$p^{10} T^{6} -$$$$20\!\cdots\!00$$$$p^{14} T^{7} +$$$$61\!\cdots\!65$$$$p^{99} T^{8} -$$$$23\!\cdots\!00$$$$p^{185} T^{9} +$$$$30\!\cdots\!41$$$$p^{271} T^{10} -$$$$44\!\cdots\!20$$$$p^{358} T^{11} +$$$$14\!\cdots\!87$$$$p^{446} T^{12} -$$$$56\!\cdots\!80$$$$p^{534} T^{13} + p^{623} T^{14}$$
23 $$1 +$$$$11\!\cdots\!04$$$$T +$$$$63\!\cdots\!73$$$$T^{2} +$$$$17\!\cdots\!80$$$$p T^{3} +$$$$16\!\cdots\!11$$$$p^{3} T^{4} +$$$$77\!\cdots\!96$$$$p^{5} T^{5} +$$$$53\!\cdots\!41$$$$p^{8} T^{6} +$$$$40\!\cdots\!60$$$$p^{11} T^{7} +$$$$53\!\cdots\!41$$$$p^{97} T^{8} +$$$$77\!\cdots\!96$$$$p^{183} T^{9} +$$$$16\!\cdots\!11$$$$p^{270} T^{10} +$$$$17\!\cdots\!80$$$$p^{357} T^{11} +$$$$63\!\cdots\!73$$$$p^{445} T^{12} +$$$$11\!\cdots\!04$$$$p^{534} T^{13} + p^{623} T^{14}$$
29 $$1 +$$$$48\!\cdots\!70$$$$p T +$$$$50\!\cdots\!63$$$$p^{2} T^{2} +$$$$56\!\cdots\!20$$$$p^{4} T^{3} +$$$$14\!\cdots\!61$$$$p^{6} T^{4} +$$$$10\!\cdots\!50$$$$p^{8} T^{5} +$$$$30\!\cdots\!15$$$$p^{10} T^{6} +$$$$20\!\cdots\!00$$$$p^{12} T^{7} +$$$$30\!\cdots\!15$$$$p^{99} T^{8} +$$$$10\!\cdots\!50$$$$p^{186} T^{9} +$$$$14\!\cdots\!61$$$$p^{273} T^{10} +$$$$56\!\cdots\!20$$$$p^{360} T^{11} +$$$$50\!\cdots\!63$$$$p^{447} T^{12} +$$$$48\!\cdots\!70$$$$p^{535} T^{13} + p^{623} T^{14}$$
31 $$1 -$$$$68\!\cdots\!04$$$$T +$$$$21\!\cdots\!61$$$$T^{2} -$$$$73\!\cdots\!64$$$$p T^{3} +$$$$21\!\cdots\!81$$$$p^{2} T^{4} -$$$$10\!\cdots\!72$$$$p^{3} T^{5} +$$$$48\!\cdots\!83$$$$p^{5} T^{6} -$$$$77\!\cdots\!52$$$$p^{7} T^{7} +$$$$48\!\cdots\!83$$$$p^{94} T^{8} -$$$$10\!\cdots\!72$$$$p^{181} T^{9} +$$$$21\!\cdots\!81$$$$p^{269} T^{10} -$$$$73\!\cdots\!64$$$$p^{357} T^{11} +$$$$21\!\cdots\!61$$$$p^{445} T^{12} -$$$$68\!\cdots\!04$$$$p^{534} T^{13} + p^{623} T^{14}$$
37 $$1 +$$$$54\!\cdots\!58$$$$T +$$$$46\!\cdots\!91$$$$p T^{2} +$$$$11\!\cdots\!80$$$$p^{2} T^{3} +$$$$28\!\cdots\!89$$$$p^{3} T^{4} +$$$$22\!\cdots\!78$$$$p^{5} T^{5} +$$$$82\!\cdots\!43$$$$p^{7} T^{6} +$$$$57\!\cdots\!80$$$$p^{9} T^{7} +$$$$82\!\cdots\!43$$$$p^{96} T^{8} +$$$$22\!\cdots\!78$$$$p^{183} T^{9} +$$$$28\!\cdots\!89$$$$p^{270} T^{10} +$$$$11\!\cdots\!80$$$$p^{358} T^{11} +$$$$46\!\cdots\!91$$$$p^{446} T^{12} +$$$$54\!\cdots\!58$$$$p^{534} T^{13} + p^{623} T^{14}$$
41 $$1 +$$$$16\!\cdots\!26$$$$p T +$$$$88\!\cdots\!71$$$$p^{2} T^{2} +$$$$16\!\cdots\!56$$$$p^{3} T^{3} +$$$$10\!\cdots\!21$$$$p^{5} T^{4} +$$$$42\!\cdots\!58$$$$p^{7} T^{5} +$$$$19\!\cdots\!63$$$$p^{9} T^{6} +$$$$64\!\cdots\!28$$$$p^{11} T^{7} +$$$$19\!\cdots\!63$$$$p^{98} T^{8} +$$$$42\!\cdots\!58$$$$p^{185} T^{9} +$$$$10\!\cdots\!21$$$$p^{272} T^{10} +$$$$16\!\cdots\!56$$$$p^{359} T^{11} +$$$$88\!\cdots\!71$$$$p^{447} T^{12} +$$$$16\!\cdots\!26$$$$p^{535} T^{13} + p^{623} T^{14}$$
43 $$1 -$$$$74\!\cdots\!92$$$$p T +$$$$54\!\cdots\!57$$$$p^{2} T^{2} -$$$$30\!\cdots\!00$$$$p^{3} T^{3} +$$$$14\!\cdots\!97$$$$p^{4} T^{4} -$$$$14\!\cdots\!68$$$$p^{6} T^{5} +$$$$13\!\cdots\!21$$$$p^{8} T^{6} -$$$$11\!\cdots\!00$$$$p^{10} T^{7} +$$$$13\!\cdots\!21$$$$p^{97} T^{8} -$$$$14\!\cdots\!68$$$$p^{184} T^{9} +$$$$14\!\cdots\!97$$$$p^{271} T^{10} -$$$$30\!\cdots\!00$$$$p^{359} T^{11} +$$$$54\!\cdots\!57$$$$p^{447} T^{12} -$$$$74\!\cdots\!92$$$$p^{535} T^{13} + p^{623} T^{14}$$
47 $$1 +$$$$58\!\cdots\!08$$$$T +$$$$50\!\cdots\!37$$$$T^{2} +$$$$21\!\cdots\!60$$$$T^{3} +$$$$22\!\cdots\!31$$$$p T^{4} +$$$$15\!\cdots\!04$$$$p^{2} T^{5} +$$$$11\!\cdots\!63$$$$p^{3} T^{6} +$$$$59\!\cdots\!80$$$$p^{4} T^{7} +$$$$11\!\cdots\!63$$$$p^{92} T^{8} +$$$$15\!\cdots\!04$$$$p^{180} T^{9} +$$$$22\!\cdots\!31$$$$p^{268} T^{10} +$$$$21\!\cdots\!60$$$$p^{356} T^{11} +$$$$50\!\cdots\!37$$$$p^{445} T^{12} +$$$$58\!\cdots\!08$$$$p^{534} T^{13} + p^{623} T^{14}$$
53 $$1 +$$$$18\!\cdots\!14$$$$T +$$$$25\!\cdots\!03$$$$T^{2} +$$$$50\!\cdots\!60$$$$p T^{3} +$$$$83\!\cdots\!53$$$$p^{2} T^{4} +$$$$11\!\cdots\!94$$$$p^{3} T^{5} +$$$$14\!\cdots\!91$$$$p^{4} T^{6} +$$$$15\!\cdots\!80$$$$p^{5} T^{7} +$$$$14\!\cdots\!91$$$$p^{93} T^{8} +$$$$11\!\cdots\!94$$$$p^{181} T^{9} +$$$$83\!\cdots\!53$$$$p^{269} T^{10} +$$$$50\!\cdots\!60$$$$p^{357} T^{11} +$$$$25\!\cdots\!03$$$$p^{445} T^{12} +$$$$18\!\cdots\!14$$$$p^{534} T^{13} + p^{623} T^{14}$$
59 $$1 +$$$$90\!\cdots\!60$$$$T +$$$$21\!\cdots\!73$$$$T^{2} +$$$$23\!\cdots\!60$$$$p T^{3} +$$$$56\!\cdots\!61$$$$p^{2} T^{4} +$$$$48\!\cdots\!00$$$$p^{3} T^{5} +$$$$92\!\cdots\!65$$$$p^{4} T^{6} +$$$$66\!\cdots\!00$$$$p^{5} T^{7} +$$$$92\!\cdots\!65$$$$p^{93} T^{8} +$$$$48\!\cdots\!00$$$$p^{181} T^{9} +$$$$56\!\cdots\!61$$$$p^{269} T^{10} +$$$$23\!\cdots\!60$$$$p^{357} T^{11} +$$$$21\!\cdots\!73$$$$p^{445} T^{12} +$$$$90\!\cdots\!60$$$$p^{534} T^{13} + p^{623} T^{14}$$
61 $$1 -$$$$87\!\cdots\!94$$$$T +$$$$29\!\cdots\!31$$$$T^{2} -$$$$40\!\cdots\!64$$$$p T^{3} +$$$$11\!\cdots\!61$$$$p^{2} T^{4} -$$$$18\!\cdots\!42$$$$p^{3} T^{5} +$$$$33\!\cdots\!43$$$$p^{4} T^{6} -$$$$51\!\cdots\!12$$$$p^{5} T^{7} +$$$$33\!\cdots\!43$$$$p^{93} T^{8} -$$$$18\!\cdots\!42$$$$p^{181} T^{9} +$$$$11\!\cdots\!61$$$$p^{269} T^{10} -$$$$40\!\cdots\!64$$$$p^{357} T^{11} +$$$$29\!\cdots\!31$$$$p^{445} T^{12} -$$$$87\!\cdots\!94$$$$p^{534} T^{13} + p^{623} T^{14}$$
67 $$1 -$$$$58\!\cdots\!92$$$$T +$$$$34\!\cdots\!77$$$$T^{2} -$$$$18\!\cdots\!80$$$$p T^{3} +$$$$90\!\cdots\!33$$$$p^{2} T^{4} -$$$$34\!\cdots\!68$$$$p^{3} T^{5} +$$$$12\!\cdots\!29$$$$p^{4} T^{6} -$$$$33\!\cdots\!40$$$$p^{5} T^{7} +$$$$12\!\cdots\!29$$$$p^{93} T^{8} -$$$$34\!\cdots\!68$$$$p^{181} T^{9} +$$$$90\!\cdots\!33$$$$p^{269} T^{10} -$$$$18\!\cdots\!80$$$$p^{357} T^{11} +$$$$34\!\cdots\!77$$$$p^{445} T^{12} -$$$$58\!\cdots\!92$$$$p^{534} T^{13} + p^{623} T^{14}$$
71 $$1 +$$$$54\!\cdots\!76$$$$T +$$$$54\!\cdots\!51$$$$p T^{2} +$$$$29\!\cdots\!16$$$$p^{2} T^{3} +$$$$17\!\cdots\!51$$$$p^{3} T^{4} +$$$$72\!\cdots\!08$$$$p^{4} T^{5} +$$$$32\!\cdots\!23$$$$p^{5} T^{6} +$$$$10\!\cdots\!48$$$$p^{6} T^{7} +$$$$32\!\cdots\!23$$$$p^{94} T^{8} +$$$$72\!\cdots\!08$$$$p^{182} T^{9} +$$$$17\!\cdots\!51$$$$p^{270} T^{10} +$$$$29\!\cdots\!16$$$$p^{358} T^{11} +$$$$54\!\cdots\!51$$$$p^{446} T^{12} +$$$$54\!\cdots\!76$$$$p^{534} T^{13} + p^{623} T^{14}$$
73 $$1 +$$$$19\!\cdots\!54$$$$T +$$$$73\!\cdots\!51$$$$p T^{2} +$$$$14\!\cdots\!60$$$$p^{2} T^{3} +$$$$30\!\cdots\!61$$$$p^{3} T^{4} +$$$$46\!\cdots\!58$$$$p^{4} T^{5} +$$$$66\!\cdots\!47$$$$p^{5} T^{6} +$$$$79\!\cdots\!80$$$$p^{6} T^{7} +$$$$66\!\cdots\!47$$$$p^{94} T^{8} +$$$$46\!\cdots\!58$$$$p^{182} T^{9} +$$$$30\!\cdots\!61$$$$p^{270} T^{10} +$$$$14\!\cdots\!60$$$$p^{358} T^{11} +$$$$73\!\cdots\!51$$$$p^{446} T^{12} +$$$$19\!\cdots\!54$$$$p^{534} T^{13} + p^{623} T^{14}$$
79 $$1 -$$$$26\!\cdots\!20$$$$T +$$$$24\!\cdots\!27$$$$p T^{2} -$$$$35\!\cdots\!80$$$$p^{2} T^{3} +$$$$28\!\cdots\!79$$$$p^{3} T^{4} -$$$$40\!\cdots\!00$$$$p^{4} T^{5} +$$$$45\!\cdots\!35$$$$p^{5} T^{6} -$$$$10\!\cdots\!00$$$$p^{6} T^{7} +$$$$45\!\cdots\!35$$$$p^{94} T^{8} -$$$$40\!\cdots\!00$$$$p^{182} T^{9} +$$$$28\!\cdots\!79$$$$p^{270} T^{10} -$$$$35\!\cdots\!80$$$$p^{358} T^{11} +$$$$24\!\cdots\!27$$$$p^{446} T^{12} -$$$$26\!\cdots\!20$$$$p^{534} T^{13} + p^{623} T^{14}$$
83 $$1 +$$$$42\!\cdots\!28$$$$p T +$$$$49\!\cdots\!97$$$$p^{2} T^{2} +$$$$14\!\cdots\!60$$$$p^{3} T^{3} +$$$$96\!\cdots\!77$$$$p^{4} T^{4} +$$$$19\!\cdots\!36$$$$p^{5} T^{5} +$$$$11\!\cdots\!09$$$$p^{6} T^{6} +$$$$19\!\cdots\!80$$$$p^{7} T^{7} +$$$$11\!\cdots\!09$$$$p^{95} T^{8} +$$$$19\!\cdots\!36$$$$p^{183} T^{9} +$$$$96\!\cdots\!77$$$$p^{271} T^{10} +$$$$14\!\cdots\!60$$$$p^{359} T^{11} +$$$$49\!\cdots\!97$$$$p^{447} T^{12} +$$$$42\!\cdots\!28$$$$p^{535} T^{13} + p^{623} T^{14}$$
89 $$1 +$$$$16\!\cdots\!90$$$$T +$$$$25\!\cdots\!63$$$$T^{2} +$$$$22\!\cdots\!60$$$$T^{3} +$$$$20\!\cdots\!01$$$$T^{4} +$$$$13\!\cdots\!50$$$$T^{5} +$$$$88\!\cdots\!15$$$$T^{6} +$$$$47\!\cdots\!00$$$$T^{7} +$$$$88\!\cdots\!15$$$$p^{89} T^{8} +$$$$13\!\cdots\!50$$$$p^{178} T^{9} +$$$$20\!\cdots\!01$$$$p^{267} T^{10} +$$$$22\!\cdots\!60$$$$p^{356} T^{11} +$$$$25\!\cdots\!63$$$$p^{445} T^{12} +$$$$16\!\cdots\!90$$$$p^{534} T^{13} + p^{623} T^{14}$$
97 $$1 -$$$$71\!\cdots\!42$$$$T +$$$$48\!\cdots\!87$$$$T^{2} -$$$$20\!\cdots\!40$$$$T^{3} +$$$$79\!\cdots\!57$$$$T^{4} -$$$$24\!\cdots\!14$$$$T^{5} +$$$$72\!\cdots\!99$$$$T^{6} -$$$$19\!\cdots\!20$$$$T^{7} +$$$$72\!\cdots\!99$$$$p^{89} T^{8} -$$$$24\!\cdots\!14$$$$p^{178} T^{9} +$$$$79\!\cdots\!57$$$$p^{267} T^{10} -$$$$20\!\cdots\!40$$$$p^{356} T^{11} +$$$$48\!\cdots\!87$$$$p^{445} T^{12} -$$$$71\!\cdots\!42$$$$p^{534} T^{13} + p^{623} T^{14}$$
show less
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}

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

−6.19462237036591940926645435560, −6.07621847382744094365184216419, −5.83807579188317149847073664613, −5.72401450927163662784671482604, −5.40996499257468715029139143785, −5.21415148062023115787017345883, −5.10416049686817180215745340848, −5.03128012492627329283061143280, −4.68605182707309679630770282207, −4.46793328136069584504190941734, −4.44123371708524621083614394267, −3.84868474186927879830908423001, −3.50641892894935739851514565752, −3.28763863480721116545096077530, −3.22121415116759283032440267067, −3.03532685951216845715275839283, −2.59148655381906762217474663683, −2.46657337848958541233786904618, −2.21905934986599851932169852405, −2.00434077516919755157598282934, −1.84015589183348535662397937312, −1.39266514465652854674679254581, −1.29402578060258513574725849796, −1.20485868071045966707975881821, −0.917234848080778442013681445998, 0, 0, 0, 0, 0, 0, 0, 0.917234848080778442013681445998, 1.20485868071045966707975881821, 1.29402578060258513574725849796, 1.39266514465652854674679254581, 1.84015589183348535662397937312, 2.00434077516919755157598282934, 2.21905934986599851932169852405, 2.46657337848958541233786904618, 2.59148655381906762217474663683, 3.03532685951216845715275839283, 3.22121415116759283032440267067, 3.28763863480721116545096077530, 3.50641892894935739851514565752, 3.84868474186927879830908423001, 4.44123371708524621083614394267, 4.46793328136069584504190941734, 4.68605182707309679630770282207, 5.03128012492627329283061143280, 5.10416049686817180215745340848, 5.21415148062023115787017345883, 5.40996499257468715029139143785, 5.72401450927163662784671482604, 5.83807579188317149847073664613, 6.07621847382744094365184216419, 6.19462237036591940926645435560

## Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.