Dirichlet series
| L(s) = 1 | + 127·2-s + 728·3-s − 1.25e4·4-s + 3.48e4·5-s + 9.24e4·6-s + 1.31e5·7-s − 2.57e6·8-s − 4.08e6·9-s + 4.43e6·10-s + 8.91e6·11-s − 9.12e6·12-s + 3.37e7·13-s + 1.67e7·14-s + 2.53e7·15-s + 2.49e7·16-s + 2.22e8·17-s − 5.18e8·18-s + 6.52e8·19-s − 4.37e8·20-s + 9.59e7·21-s + 1.13e9·22-s + 1.79e9·23-s − 1.87e9·24-s − 3.54e9·25-s + 4.29e9·26-s − 4.12e9·27-s − 1.65e9·28-s + ⋯ |
| L(s) = 1 | + 1.40·2-s + 0.576·3-s − 1.52·4-s + 0.998·5-s + 0.809·6-s + 0.423·7-s − 3.47·8-s − 2.56·9-s + 1.40·10-s + 1.51·11-s − 0.881·12-s + 1.94·13-s + 0.594·14-s + 0.575·15-s + 0.371·16-s + 2.23·17-s − 3.59·18-s + 3.18·19-s − 1.52·20-s + 0.244·21-s + 2.12·22-s + 2.52·23-s − 2.00·24-s − 2.90·25-s + 2.72·26-s − 2.04·27-s − 0.648·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 62748517 ^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62748517 ^{s/2} \, \Gamma_{\C}(s+13/2)^{7} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(62748517\) = \(13^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.02292\times 10^{8}\) |
| Root analytic conductor: | \(3.73363\) |
| Motivic weight: | \(13\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 62748517,\ (\ :[13/2]^{7}),\ 1)\) |
Particular Values
| \(L(7)\) | \(\approx\) | \(12.24739819\) |
| \(L(\frac12)\) | \(\approx\) | \(12.24739819\) |
| \(L(\frac{15}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 13 | \( ( 1 - p^{6} T )^{7} \) |
| good | 2 | \( 1 - 127 T + 7165 p^{2} T^{2} - 663197 p^{2} T^{3} + 10739125 p^{5} T^{4} - 328336345 p^{6} T^{5} + 302977677 p^{13} T^{6} - 7767365167 p^{14} T^{7} + 302977677 p^{26} T^{8} - 328336345 p^{32} T^{9} + 10739125 p^{44} T^{10} - 663197 p^{54} T^{11} + 7165 p^{67} T^{12} - 127 p^{78} T^{13} + p^{91} T^{14} \) |
| 3 | \( 1 - 728 T + 512455 p^{2} T^{2} - 244830004 p^{2} T^{3} + 104914535960 p^{4} T^{4} + 1229369454692 p^{6} T^{5} + 173234292383762 p^{10} T^{6} + 15346048309704112 p^{12} T^{7} + 173234292383762 p^{23} T^{8} + 1229369454692 p^{32} T^{9} + 104914535960 p^{43} T^{10} - 244830004 p^{54} T^{11} + 512455 p^{67} T^{12} - 728 p^{78} T^{13} + p^{91} T^{14} \) | |
| 5 | \( 1 - 34886 T + 4757900029 T^{2} - 156968742782524 T^{3} + 2438538641649432136 p T^{4} - \)\(56\!\cdots\!16\)\( p^{4} T^{5} + \)\(67\!\cdots\!14\)\( p^{5} T^{6} - \)\(66\!\cdots\!96\)\( p^{7} T^{7} + \)\(67\!\cdots\!14\)\( p^{18} T^{8} - \)\(56\!\cdots\!16\)\( p^{30} T^{9} + 2438538641649432136 p^{40} T^{10} - 156968742782524 p^{52} T^{11} + 4757900029 p^{65} T^{12} - 34886 p^{78} T^{13} + p^{91} T^{14} \) | |
| 7 | \( 1 - 131864 T + 45729160957 p T^{2} - 57434191382362972 T^{3} + \)\(40\!\cdots\!00\)\( T^{4} - \)\(19\!\cdots\!68\)\( p^{2} T^{5} + \)\(61\!\cdots\!66\)\( p^{2} T^{6} - \)\(30\!\cdots\!96\)\( p^{3} T^{7} + \)\(61\!\cdots\!66\)\( p^{15} T^{8} - \)\(19\!\cdots\!68\)\( p^{28} T^{9} + \)\(40\!\cdots\!00\)\( p^{39} T^{10} - 57434191382362972 p^{52} T^{11} + 45729160957 p^{66} T^{12} - 131864 p^{78} T^{13} + p^{91} T^{14} \) | |
| 11 | \( 1 - 8912632 T + 158451197097829 T^{2} - \)\(11\!\cdots\!00\)\( p T^{3} + \)\(10\!\cdots\!57\)\( p^{2} T^{4} - \)\(63\!\cdots\!60\)\( p^{3} T^{5} + \)\(45\!\cdots\!05\)\( p^{4} T^{6} - \)\(22\!\cdots\!60\)\( p^{5} T^{7} + \)\(45\!\cdots\!05\)\( p^{17} T^{8} - \)\(63\!\cdots\!60\)\( p^{29} T^{9} + \)\(10\!\cdots\!57\)\( p^{41} T^{10} - \)\(11\!\cdots\!00\)\( p^{53} T^{11} + 158451197097829 p^{65} T^{12} - 8912632 p^{78} T^{13} + p^{91} T^{14} \) | |
| 17 | \( 1 - 222199706 T + 60872086724599249 T^{2} - \)\(93\!\cdots\!36\)\( T^{3} + \)\(14\!\cdots\!60\)\( T^{4} - \)\(17\!\cdots\!72\)\( T^{5} + \)\(12\!\cdots\!58\)\( p T^{6} - \)\(21\!\cdots\!84\)\( T^{7} + \)\(12\!\cdots\!58\)\( p^{14} T^{8} - \)\(17\!\cdots\!72\)\( p^{26} T^{9} + \)\(14\!\cdots\!60\)\( p^{39} T^{10} - \)\(93\!\cdots\!36\)\( p^{52} T^{11} + 60872086724599249 p^{65} T^{12} - 222199706 p^{78} T^{13} + p^{91} T^{14} \) | |
| 19 | \( 1 - 652665968 T + 445931393424331117 T^{2} - \)\(17\!\cdots\!40\)\( T^{3} + \)\(69\!\cdots\!65\)\( T^{4} - \)\(19\!\cdots\!72\)\( T^{5} + \)\(53\!\cdots\!89\)\( T^{6} - \)\(11\!\cdots\!00\)\( T^{7} + \)\(53\!\cdots\!89\)\( p^{13} T^{8} - \)\(19\!\cdots\!72\)\( p^{26} T^{9} + \)\(69\!\cdots\!65\)\( p^{39} T^{10} - \)\(17\!\cdots\!40\)\( p^{52} T^{11} + 445931393424331117 p^{65} T^{12} - 652665968 p^{78} T^{13} + p^{91} T^{14} \) | |
| 23 | \( 1 - 1794723984 T + 2893038968226440705 T^{2} - \)\(31\!\cdots\!56\)\( T^{3} + \)\(32\!\cdots\!45\)\( T^{4} - \)\(28\!\cdots\!28\)\( T^{5} + \)\(23\!\cdots\!33\)\( T^{6} - \)\(17\!\cdots\!08\)\( T^{7} + \)\(23\!\cdots\!33\)\( p^{13} T^{8} - \)\(28\!\cdots\!28\)\( p^{26} T^{9} + \)\(32\!\cdots\!45\)\( p^{39} T^{10} - \)\(31\!\cdots\!56\)\( p^{52} T^{11} + 2893038968226440705 p^{65} T^{12} - 1794723984 p^{78} T^{13} + p^{91} T^{14} \) | |
| 29 | \( 1 + 4735865782 T + 19337064041122538767 T^{2} - \)\(24\!\cdots\!56\)\( T^{3} - \)\(41\!\cdots\!67\)\( T^{4} - \)\(28\!\cdots\!74\)\( T^{5} + \)\(21\!\cdots\!31\)\( T^{6} + \)\(65\!\cdots\!36\)\( T^{7} + \)\(21\!\cdots\!31\)\( p^{13} T^{8} - \)\(28\!\cdots\!74\)\( p^{26} T^{9} - \)\(41\!\cdots\!67\)\( p^{39} T^{10} - \)\(24\!\cdots\!56\)\( p^{52} T^{11} + 19337064041122538767 p^{65} T^{12} + 4735865782 p^{78} T^{13} + p^{91} T^{14} \) | |
| 31 | \( 1 + 9436127356 T + 72108590323411014649 T^{2} + \)\(47\!\cdots\!24\)\( T^{3} + \)\(34\!\cdots\!65\)\( T^{4} + \)\(21\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!53\)\( T^{6} + \)\(54\!\cdots\!64\)\( T^{7} + \)\(11\!\cdots\!53\)\( p^{13} T^{8} + \)\(21\!\cdots\!00\)\( p^{26} T^{9} + \)\(34\!\cdots\!65\)\( p^{39} T^{10} + \)\(47\!\cdots\!24\)\( p^{52} T^{11} + 72108590323411014649 p^{65} T^{12} + 9436127356 p^{78} T^{13} + p^{91} T^{14} \) | |
| 37 | \( 1 + 14832967330 T + \)\(97\!\cdots\!25\)\( T^{2} + \)\(17\!\cdots\!72\)\( T^{3} + \)\(50\!\cdots\!60\)\( T^{4} + \)\(96\!\cdots\!24\)\( T^{5} + \)\(17\!\cdots\!22\)\( T^{6} + \)\(30\!\cdots\!96\)\( T^{7} + \)\(17\!\cdots\!22\)\( p^{13} T^{8} + \)\(96\!\cdots\!24\)\( p^{26} T^{9} + \)\(50\!\cdots\!60\)\( p^{39} T^{10} + \)\(17\!\cdots\!72\)\( p^{52} T^{11} + \)\(97\!\cdots\!25\)\( p^{65} T^{12} + 14832967330 p^{78} T^{13} + p^{91} T^{14} \) | |
| 41 | \( 1 - 59907995430 T + \)\(43\!\cdots\!03\)\( T^{2} - \)\(16\!\cdots\!76\)\( T^{3} + \)\(83\!\cdots\!81\)\( T^{4} - \)\(26\!\cdots\!46\)\( T^{5} + \)\(10\!\cdots\!03\)\( T^{6} - \)\(29\!\cdots\!92\)\( T^{7} + \)\(10\!\cdots\!03\)\( p^{13} T^{8} - \)\(26\!\cdots\!46\)\( p^{26} T^{9} + \)\(83\!\cdots\!81\)\( p^{39} T^{10} - \)\(16\!\cdots\!76\)\( p^{52} T^{11} + \)\(43\!\cdots\!03\)\( p^{65} T^{12} - 59907995430 p^{78} T^{13} + p^{91} T^{14} \) | |
| 43 | \( 1 - 72489466184 T + \)\(90\!\cdots\!31\)\( T^{2} - \)\(41\!\cdots\!60\)\( T^{3} + \)\(31\!\cdots\!88\)\( T^{4} - \)\(23\!\cdots\!52\)\( p T^{5} + \)\(63\!\cdots\!98\)\( T^{6} - \)\(17\!\cdots\!92\)\( T^{7} + \)\(63\!\cdots\!98\)\( p^{13} T^{8} - \)\(23\!\cdots\!52\)\( p^{27} T^{9} + \)\(31\!\cdots\!88\)\( p^{39} T^{10} - \)\(41\!\cdots\!60\)\( p^{52} T^{11} + \)\(90\!\cdots\!31\)\( p^{65} T^{12} - 72489466184 p^{78} T^{13} + p^{91} T^{14} \) | |
| 47 | \( 1 - 195641345592 T + \)\(28\!\cdots\!87\)\( T^{2} - \)\(24\!\cdots\!04\)\( T^{3} + \)\(17\!\cdots\!72\)\( T^{4} - \)\(68\!\cdots\!72\)\( T^{5} + \)\(25\!\cdots\!42\)\( T^{6} - \)\(23\!\cdots\!48\)\( T^{7} + \)\(25\!\cdots\!42\)\( p^{13} T^{8} - \)\(68\!\cdots\!72\)\( p^{26} T^{9} + \)\(17\!\cdots\!72\)\( p^{39} T^{10} - \)\(24\!\cdots\!04\)\( p^{52} T^{11} + \)\(28\!\cdots\!87\)\( p^{65} T^{12} - 195641345592 p^{78} T^{13} + p^{91} T^{14} \) | |
| 53 | \( 1 - 409286178474 T + \)\(15\!\cdots\!39\)\( T^{2} - \)\(39\!\cdots\!96\)\( T^{3} + \)\(94\!\cdots\!77\)\( T^{4} - \)\(19\!\cdots\!02\)\( T^{5} + \)\(36\!\cdots\!95\)\( T^{6} - \)\(62\!\cdots\!68\)\( T^{7} + \)\(36\!\cdots\!95\)\( p^{13} T^{8} - \)\(19\!\cdots\!02\)\( p^{26} T^{9} + \)\(94\!\cdots\!77\)\( p^{39} T^{10} - \)\(39\!\cdots\!96\)\( p^{52} T^{11} + \)\(15\!\cdots\!39\)\( p^{65} T^{12} - 409286178474 p^{78} T^{13} + p^{91} T^{14} \) | |
| 59 | \( 1 + 160361676600 T + \)\(50\!\cdots\!49\)\( T^{2} + \)\(60\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!13\)\( T^{4} + \)\(12\!\cdots\!80\)\( T^{5} + \)\(19\!\cdots\!65\)\( T^{6} + \)\(16\!\cdots\!80\)\( T^{7} + \)\(19\!\cdots\!65\)\( p^{13} T^{8} + \)\(12\!\cdots\!80\)\( p^{26} T^{9} + \)\(12\!\cdots\!13\)\( p^{39} T^{10} + \)\(60\!\cdots\!20\)\( p^{52} T^{11} + \)\(50\!\cdots\!49\)\( p^{65} T^{12} + 160361676600 p^{78} T^{13} + p^{91} T^{14} \) | |
| 61 | \( 1 + 147184998046 T + \)\(49\!\cdots\!43\)\( T^{2} + \)\(16\!\cdots\!64\)\( T^{3} + \)\(15\!\cdots\!17\)\( T^{4} + \)\(34\!\cdots\!62\)\( T^{5} + \)\(33\!\cdots\!95\)\( T^{6} + \)\(11\!\cdots\!24\)\( T^{7} + \)\(33\!\cdots\!95\)\( p^{13} T^{8} + \)\(34\!\cdots\!62\)\( p^{26} T^{9} + \)\(15\!\cdots\!17\)\( p^{39} T^{10} + \)\(16\!\cdots\!64\)\( p^{52} T^{11} + \)\(49\!\cdots\!43\)\( p^{65} T^{12} + 147184998046 p^{78} T^{13} + p^{91} T^{14} \) | |
| 67 | \( 1 - 1533110763584 T + \)\(27\!\cdots\!33\)\( T^{2} - \)\(23\!\cdots\!48\)\( T^{3} + \)\(24\!\cdots\!21\)\( T^{4} - \)\(13\!\cdots\!84\)\( T^{5} + \)\(12\!\cdots\!81\)\( T^{6} - \)\(61\!\cdots\!64\)\( T^{7} + \)\(12\!\cdots\!81\)\( p^{13} T^{8} - \)\(13\!\cdots\!84\)\( p^{26} T^{9} + \)\(24\!\cdots\!21\)\( p^{39} T^{10} - \)\(23\!\cdots\!48\)\( p^{52} T^{11} + \)\(27\!\cdots\!33\)\( p^{65} T^{12} - 1533110763584 p^{78} T^{13} + p^{91} T^{14} \) | |
| 71 | \( 1 - 1288841132520 T + \)\(56\!\cdots\!75\)\( T^{2} - \)\(77\!\cdots\!04\)\( T^{3} + \)\(16\!\cdots\!64\)\( T^{4} - \)\(19\!\cdots\!84\)\( T^{5} + \)\(28\!\cdots\!22\)\( T^{6} - \)\(29\!\cdots\!48\)\( T^{7} + \)\(28\!\cdots\!22\)\( p^{13} T^{8} - \)\(19\!\cdots\!84\)\( p^{26} T^{9} + \)\(16\!\cdots\!64\)\( p^{39} T^{10} - \)\(77\!\cdots\!04\)\( p^{52} T^{11} + \)\(56\!\cdots\!75\)\( p^{65} T^{12} - 1288841132520 p^{78} T^{13} + p^{91} T^{14} \) | |
| 73 | \( 1 - 4340518042046 T + \)\(15\!\cdots\!87\)\( T^{2} - \)\(35\!\cdots\!24\)\( T^{3} + \)\(76\!\cdots\!49\)\( T^{4} - \)\(13\!\cdots\!82\)\( T^{5} + \)\(21\!\cdots\!23\)\( T^{6} - \)\(28\!\cdots\!72\)\( T^{7} + \)\(21\!\cdots\!23\)\( p^{13} T^{8} - \)\(13\!\cdots\!82\)\( p^{26} T^{9} + \)\(76\!\cdots\!49\)\( p^{39} T^{10} - \)\(35\!\cdots\!24\)\( p^{52} T^{11} + \)\(15\!\cdots\!87\)\( p^{65} T^{12} - 4340518042046 p^{78} T^{13} + p^{91} T^{14} \) | |
| 79 | \( 1 - 6986884509272 T + \)\(42\!\cdots\!57\)\( T^{2} - \)\(17\!\cdots\!32\)\( T^{3} + \)\(64\!\cdots\!17\)\( T^{4} - \)\(19\!\cdots\!88\)\( T^{5} + \)\(51\!\cdots\!41\)\( T^{6} - \)\(11\!\cdots\!36\)\( T^{7} + \)\(51\!\cdots\!41\)\( p^{13} T^{8} - \)\(19\!\cdots\!88\)\( p^{26} T^{9} + \)\(64\!\cdots\!17\)\( p^{39} T^{10} - \)\(17\!\cdots\!32\)\( p^{52} T^{11} + \)\(42\!\cdots\!57\)\( p^{65} T^{12} - 6986884509272 p^{78} T^{13} + p^{91} T^{14} \) | |
| 83 | \( 1 - 9485774126680 T + \)\(84\!\cdots\!81\)\( T^{2} - \)\(46\!\cdots\!68\)\( T^{3} + \)\(24\!\cdots\!77\)\( T^{4} - \)\(97\!\cdots\!96\)\( T^{5} + \)\(36\!\cdots\!33\)\( T^{6} - \)\(11\!\cdots\!24\)\( T^{7} + \)\(36\!\cdots\!33\)\( p^{13} T^{8} - \)\(97\!\cdots\!96\)\( p^{26} T^{9} + \)\(24\!\cdots\!77\)\( p^{39} T^{10} - \)\(46\!\cdots\!68\)\( p^{52} T^{11} + \)\(84\!\cdots\!81\)\( p^{65} T^{12} - 9485774126680 p^{78} T^{13} + p^{91} T^{14} \) | |
| 89 | \( 1 + 1338832432586 T + \)\(54\!\cdots\!11\)\( T^{2} + \)\(28\!\cdots\!60\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} + \)\(10\!\cdots\!90\)\( T^{5} + \)\(73\!\cdots\!55\)\( T^{6} + \)\(22\!\cdots\!48\)\( T^{7} + \)\(73\!\cdots\!55\)\( p^{13} T^{8} + \)\(10\!\cdots\!90\)\( p^{26} T^{9} + \)\(19\!\cdots\!41\)\( p^{39} T^{10} + \)\(28\!\cdots\!60\)\( p^{52} T^{11} + \)\(54\!\cdots\!11\)\( p^{65} T^{12} + 1338832432586 p^{78} T^{13} + p^{91} T^{14} \) | |
| 97 | \( 1 - 1532637634742 T + \)\(25\!\cdots\!59\)\( T^{2} - \)\(10\!\cdots\!04\)\( T^{3} + \)\(35\!\cdots\!05\)\( T^{4} - \)\(14\!\cdots\!38\)\( T^{5} + \)\(34\!\cdots\!03\)\( T^{6} - \)\(12\!\cdots\!12\)\( T^{7} + \)\(34\!\cdots\!03\)\( p^{13} T^{8} - \)\(14\!\cdots\!38\)\( p^{26} T^{9} + \)\(35\!\cdots\!05\)\( p^{39} T^{10} - \)\(10\!\cdots\!04\)\( p^{52} T^{11} + \)\(25\!\cdots\!59\)\( p^{65} T^{12} - 1532637634742 p^{78} T^{13} + p^{91} T^{14} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.55526288924803430480599310697, −7.41896479898108487636740268179, −6.58666851326403773994966614505, −6.41334982560676940217496229120, −6.17636354104737005654745643586, −5.60552004659410810748613671342, −5.59876597396294778306212505117, −5.48356842451459580464218652194, −5.41326735357838008548895737218, −5.13230884435209228378939777973, −4.93717470424471505184398066360, −4.14844593626068139561079908676, −3.82456065062436281402358718360, −3.66148541241920141934256379066, −3.55528473979559362000336088663, −3.54682371286425068416346117259, −3.20517724287271897234234825377, −2.66320342127227645143445773737, −2.24994594379725112091786086748, −1.97323331217942024065385148956, −1.48954968652076895346605183296, −0.924205832651695997872054234125, −0.891139499494482286924277930957, −0.809039669674081391395557073164, −0.26494562157779490518252433396, 0.26494562157779490518252433396, 0.809039669674081391395557073164, 0.891139499494482286924277930957, 0.924205832651695997872054234125, 1.48954968652076895346605183296, 1.97323331217942024065385148956, 2.24994594379725112091786086748, 2.66320342127227645143445773737, 3.20517724287271897234234825377, 3.54682371286425068416346117259, 3.55528473979559362000336088663, 3.66148541241920141934256379066, 3.82456065062436281402358718360, 4.14844593626068139561079908676, 4.93717470424471505184398066360, 5.13230884435209228378939777973, 5.41326735357838008548895737218, 5.48356842451459580464218652194, 5.59876597396294778306212505117, 5.60552004659410810748613671342, 6.17636354104737005654745643586, 6.41334982560676940217496229120, 6.58666851326403773994966614505, 7.41896479898108487636740268179, 7.55526288924803430480599310697
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.