Dirichlet series
| L(s) = 1 | − 64·2-s + 480·3-s − 1.96e4·4-s − 454·5-s − 3.07e4·6-s − 3.13e5·7-s + 1.43e6·8-s − 3.20e6·9-s + 2.90e4·10-s − 8.85e6·11-s − 9.42e6·12-s − 3.63e7·13-s + 2.00e7·14-s − 2.17e5·15-s + 1.58e8·16-s − 3.09e8·17-s + 2.05e8·18-s − 1.47e8·19-s + 8.91e6·20-s − 1.50e8·21-s + 5.66e8·22-s + 6.77e8·23-s + 6.88e8·24-s − 2.28e9·25-s + 2.32e9·26-s − 1.55e8·27-s + 6.16e9·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.380·3-s − 2.39·4-s − 0.0129·5-s − 0.268·6-s − 1.00·7-s + 1.93·8-s − 2.01·9-s + 0.00918·10-s − 1.50·11-s − 0.911·12-s − 2.08·13-s + 0.713·14-s − 0.00493·15-s + 2.35·16-s − 3.10·17-s + 1.42·18-s − 0.717·19-s + 0.0311·20-s − 0.383·21-s + 1.06·22-s + 0.954·23-s + 0.734·24-s − 1.86·25-s + 1.47·26-s − 0.0774·27-s + 2.41·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 161051 ^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161051 ^{s/2} \, \Gamma_{\C}(s+13/2)^{5} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(10\) |
| Conductor: | \(161051\) = \(11^{5}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(228330.\) |
| Root analytic conductor: | \(3.43444\) |
| Motivic weight: | \(13\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(5\) |
| Selberg data: | \((10,\ 161051,\ (\ :13/2, 13/2, 13/2, 13/2, 13/2),\ -1)\) |
Particular Values
| \(L(7)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{15}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| bad | 11 | $C_1$ | \( ( 1 + p^{6} T )^{5} \) |
| good | 2 | $C_2 \wr S_5$ | \( 1 + p^{6} T + 1483 p^{4} T^{2} + 20961 p^{6} T^{3} + 1178477 p^{8} T^{4} + 6442325 p^{11} T^{5} + 1178477 p^{21} T^{6} + 20961 p^{32} T^{7} + 1483 p^{43} T^{8} + p^{58} T^{9} + p^{65} T^{10} \) |
| 3 | $C_2 \wr S_5$ | \( 1 - 160 p T + 382237 p^{2} T^{2} - 37480700 p^{4} T^{3} + 929623921 p^{8} T^{4} - 111114137620 p^{10} T^{5} + 929623921 p^{21} T^{6} - 37480700 p^{30} T^{7} + 382237 p^{41} T^{8} - 160 p^{53} T^{9} + p^{65} T^{10} \) | |
| 5 | $C_2 \wr S_5$ | \( 1 + 454 T + 456345611 p T^{2} - 2089728239616 p^{2} T^{3} + 5267172313686821 p^{4} T^{4} - 5802162350535077666 p^{6} T^{5} + 5267172313686821 p^{17} T^{6} - 2089728239616 p^{28} T^{7} + 456345611 p^{40} T^{8} + 454 p^{52} T^{9} + p^{65} T^{10} \) | |
| 7 | $C_2 \wr S_5$ | \( 1 + 313920 T + 36080224813 p T^{2} + 71177329102381280 T^{3} + \)\(30\!\cdots\!50\)\( T^{4} + \)\(11\!\cdots\!20\)\( p T^{5} + \)\(30\!\cdots\!50\)\( p^{13} T^{6} + 71177329102381280 p^{26} T^{7} + 36080224813 p^{40} T^{8} + 313920 p^{52} T^{9} + p^{65} T^{10} \) | |
| 13 | $C_2 \wr S_5$ | \( 1 + 2795346 p T + 1235982606237265 T^{2} + \)\(30\!\cdots\!76\)\( T^{3} + \)\(71\!\cdots\!90\)\( T^{4} + \)\(12\!\cdots\!92\)\( T^{5} + \)\(71\!\cdots\!90\)\( p^{13} T^{6} + \)\(30\!\cdots\!76\)\( p^{26} T^{7} + 1235982606237265 p^{39} T^{8} + 2795346 p^{53} T^{9} + p^{65} T^{10} \) | |
| 17 | $C_2 \wr S_5$ | \( 1 + 309078454 T + 80219506365980317 T^{2} + \)\(13\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!46\)\( T^{4} + \)\(20\!\cdots\!44\)\( T^{5} + \)\(18\!\cdots\!46\)\( p^{13} T^{6} + \)\(13\!\cdots\!20\)\( p^{26} T^{7} + 80219506365980317 p^{39} T^{8} + 309078454 p^{52} T^{9} + p^{65} T^{10} \) | |
| 19 | $C_2 \wr S_5$ | \( 1 + 147232948 T + 134129791315111727 T^{2} + \)\(14\!\cdots\!96\)\( T^{3} + \)\(49\!\cdots\!54\)\( p T^{4} + \)\(84\!\cdots\!96\)\( T^{5} + \)\(49\!\cdots\!54\)\( p^{14} T^{6} + \)\(14\!\cdots\!96\)\( p^{26} T^{7} + 134129791315111727 p^{39} T^{8} + 147232948 p^{52} T^{9} + p^{65} T^{10} \) | |
| 23 | $C_2 \wr S_5$ | \( 1 - 677905444 T + 1839977437563311353 T^{2} - \)\(94\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!81\)\( T^{4} - \)\(60\!\cdots\!24\)\( T^{5} + \)\(14\!\cdots\!81\)\( p^{13} T^{6} - \)\(94\!\cdots\!80\)\( p^{26} T^{7} + 1839977437563311353 p^{39} T^{8} - 677905444 p^{52} T^{9} + p^{65} T^{10} \) | |
| 29 | $C_2 \wr S_5$ | \( 1 - 2368825878 T + 16504996339998686785 T^{2} - \)\(51\!\cdots\!08\)\( T^{3} + \)\(14\!\cdots\!78\)\( T^{4} - \)\(47\!\cdots\!24\)\( T^{5} + \)\(14\!\cdots\!78\)\( p^{13} T^{6} - \)\(51\!\cdots\!08\)\( p^{26} T^{7} + 16504996339998686785 p^{39} T^{8} - 2368825878 p^{52} T^{9} + p^{65} T^{10} \) | |
| 31 | $C_2 \wr S_5$ | \( 1 + 83363076 T + 41639952015475292929 T^{2} - \)\(98\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!89\)\( T^{4} - \)\(32\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!89\)\( p^{13} T^{6} - \)\(98\!\cdots\!60\)\( p^{26} T^{7} + 41639952015475292929 p^{39} T^{8} + 83363076 p^{52} T^{9} + p^{65} T^{10} \) | |
| 37 | $C_2 \wr S_5$ | \( 1 + 32935650382 T + \)\(11\!\cdots\!87\)\( T^{2} + \)\(27\!\cdots\!80\)\( T^{3} + \)\(56\!\cdots\!01\)\( T^{4} + \)\(96\!\cdots\!22\)\( T^{5} + \)\(56\!\cdots\!01\)\( p^{13} T^{6} + \)\(27\!\cdots\!80\)\( p^{26} T^{7} + \)\(11\!\cdots\!87\)\( p^{39} T^{8} + 32935650382 p^{52} T^{9} + p^{65} T^{10} \) | |
| 41 | $C_2 \wr S_5$ | \( 1 + 70273827286 T + \)\(64\!\cdots\!57\)\( T^{2} + \)\(28\!\cdots\!36\)\( T^{3} + \)\(13\!\cdots\!86\)\( T^{4} + \)\(39\!\cdots\!48\)\( T^{5} + \)\(13\!\cdots\!86\)\( p^{13} T^{6} + \)\(28\!\cdots\!36\)\( p^{26} T^{7} + \)\(64\!\cdots\!57\)\( p^{39} T^{8} + 70273827286 p^{52} T^{9} + p^{65} T^{10} \) | |
| 43 | $C_2 \wr S_5$ | \( 1 + 54501240436 T + \)\(48\!\cdots\!51\)\( T^{2} + \)\(15\!\cdots\!88\)\( T^{3} + \)\(12\!\cdots\!58\)\( T^{4} + \)\(36\!\cdots\!04\)\( T^{5} + \)\(12\!\cdots\!58\)\( p^{13} T^{6} + \)\(15\!\cdots\!88\)\( p^{26} T^{7} + \)\(48\!\cdots\!51\)\( p^{39} T^{8} + 54501240436 p^{52} T^{9} + p^{65} T^{10} \) | |
| 47 | $C_2 \wr S_5$ | \( 1 + 45017434472 T + \)\(19\!\cdots\!07\)\( T^{2} + \)\(75\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!18\)\( T^{4} + \)\(55\!\cdots\!76\)\( T^{5} + \)\(16\!\cdots\!18\)\( p^{13} T^{6} + \)\(75\!\cdots\!60\)\( p^{26} T^{7} + \)\(19\!\cdots\!07\)\( p^{39} T^{8} + 45017434472 p^{52} T^{9} + p^{65} T^{10} \) | |
| 53 | $C_2 \wr S_5$ | \( 1 - 242684257518 T + \)\(96\!\cdots\!13\)\( T^{2} - \)\(32\!\cdots\!40\)\( p T^{3} + \)\(43\!\cdots\!46\)\( T^{4} - \)\(62\!\cdots\!68\)\( T^{5} + \)\(43\!\cdots\!46\)\( p^{13} T^{6} - \)\(32\!\cdots\!40\)\( p^{27} T^{7} + \)\(96\!\cdots\!13\)\( p^{39} T^{8} - 242684257518 p^{52} T^{9} + p^{65} T^{10} \) | |
| 59 | $C_2 \wr S_5$ | \( 1 + 384712501184 T + \)\(43\!\cdots\!33\)\( T^{2} + \)\(11\!\cdots\!40\)\( T^{3} + \)\(81\!\cdots\!61\)\( T^{4} + \)\(17\!\cdots\!76\)\( T^{5} + \)\(81\!\cdots\!61\)\( p^{13} T^{6} + \)\(11\!\cdots\!40\)\( p^{26} T^{7} + \)\(43\!\cdots\!33\)\( p^{39} T^{8} + 384712501184 p^{52} T^{9} + p^{65} T^{10} \) | |
| 61 | $C_2 \wr S_5$ | \( 1 + 795317095690 T + \)\(87\!\cdots\!49\)\( T^{2} + \)\(47\!\cdots\!80\)\( T^{3} + \)\(28\!\cdots\!86\)\( T^{4} + \)\(11\!\cdots\!60\)\( T^{5} + \)\(28\!\cdots\!86\)\( p^{13} T^{6} + \)\(47\!\cdots\!80\)\( p^{26} T^{7} + \)\(87\!\cdots\!49\)\( p^{39} T^{8} + 795317095690 p^{52} T^{9} + p^{65} T^{10} \) | |
| 67 | $C_2 \wr S_5$ | \( 1 + 1005952134296 T + \)\(16\!\cdots\!93\)\( T^{2} + \)\(11\!\cdots\!76\)\( T^{3} + \)\(14\!\cdots\!69\)\( T^{4} + \)\(93\!\cdots\!12\)\( T^{5} + \)\(14\!\cdots\!69\)\( p^{13} T^{6} + \)\(11\!\cdots\!76\)\( p^{26} T^{7} + \)\(16\!\cdots\!93\)\( p^{39} T^{8} + 1005952134296 p^{52} T^{9} + p^{65} T^{10} \) | |
| 71 | $C_2 \wr S_5$ | \( 1 + 1427050574148 T + \)\(43\!\cdots\!17\)\( T^{2} + \)\(47\!\cdots\!40\)\( T^{3} + \)\(86\!\cdots\!41\)\( T^{4} + \)\(71\!\cdots\!52\)\( T^{5} + \)\(86\!\cdots\!41\)\( p^{13} T^{6} + \)\(47\!\cdots\!40\)\( p^{26} T^{7} + \)\(43\!\cdots\!17\)\( p^{39} T^{8} + 1427050574148 p^{52} T^{9} + p^{65} T^{10} \) | |
| 73 | $C_2 \wr S_5$ | \( 1 + 4111049036406 T + \)\(10\!\cdots\!29\)\( T^{2} + \)\(20\!\cdots\!76\)\( T^{3} + \)\(35\!\cdots\!50\)\( T^{4} + \)\(48\!\cdots\!92\)\( T^{5} + \)\(35\!\cdots\!50\)\( p^{13} T^{6} + \)\(20\!\cdots\!76\)\( p^{26} T^{7} + \)\(10\!\cdots\!29\)\( p^{39} T^{8} + 4111049036406 p^{52} T^{9} + p^{65} T^{10} \) | |
| 79 | $C_2 \wr S_5$ | \( 1 + 3666957194024 T + \)\(77\!\cdots\!27\)\( T^{2} + \)\(52\!\cdots\!32\)\( T^{3} - \)\(63\!\cdots\!06\)\( T^{4} - \)\(49\!\cdots\!80\)\( T^{5} - \)\(63\!\cdots\!06\)\( p^{13} T^{6} + \)\(52\!\cdots\!32\)\( p^{26} T^{7} + \)\(77\!\cdots\!27\)\( p^{39} T^{8} + 3666957194024 p^{52} T^{9} + p^{65} T^{10} \) | |
| 83 | $C_2 \wr S_5$ | \( 1 - 2718055516116 T + \)\(29\!\cdots\!47\)\( T^{2} - \)\(80\!\cdots\!44\)\( T^{3} + \)\(41\!\cdots\!94\)\( T^{4} - \)\(10\!\cdots\!32\)\( T^{5} + \)\(41\!\cdots\!94\)\( p^{13} T^{6} - \)\(80\!\cdots\!44\)\( p^{26} T^{7} + \)\(29\!\cdots\!47\)\( p^{39} T^{8} - 2718055516116 p^{52} T^{9} + p^{65} T^{10} \) | |
| 89 | $C_2 \wr S_5$ | \( 1 - 7963494884214 T + \)\(54\!\cdots\!27\)\( T^{2} - \)\(12\!\cdots\!20\)\( T^{3} - \)\(91\!\cdots\!75\)\( T^{4} + \)\(26\!\cdots\!98\)\( T^{5} - \)\(91\!\cdots\!75\)\( p^{13} T^{6} - \)\(12\!\cdots\!20\)\( p^{26} T^{7} + \)\(54\!\cdots\!27\)\( p^{39} T^{8} - 7963494884214 p^{52} T^{9} + p^{65} T^{10} \) | |
| 97 | $C_2 \wr S_5$ | \( 1 + 13542719272730 T + \)\(35\!\cdots\!39\)\( T^{2} + \)\(33\!\cdots\!60\)\( T^{3} + \)\(48\!\cdots\!65\)\( T^{4} + \)\(32\!\cdots\!30\)\( T^{5} + \)\(48\!\cdots\!65\)\( p^{13} T^{6} + \)\(33\!\cdots\!60\)\( p^{26} T^{7} + \)\(35\!\cdots\!39\)\( p^{39} T^{8} + 13542719272730 p^{52} T^{9} + p^{65} T^{10} \) | |
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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
−11.12167393245733668684114048203, −10.45334061861574183526918956826, −10.16156449821809432477882806299, −10.11522513621733920078992331813, −9.758975601402302747241374336491, −9.113537941465847528066041821250, −8.968669119891794363502089843787, −8.804482240001268229157234513930, −8.493811020560654924164991290973, −8.444598562801082828116530343262, −7.77930575353127022717719805558, −7.38404150983480921923092180577, −6.93469958466916536339152157830, −6.30061561179429969377654018036, −6.22045722891093812723762630188, −5.29342614602785183353818983280, −5.11673631804307418603723291353, −4.68295572456568848726380961980, −4.63211455163064979737899825903, −4.04555556760300556346583373584, −3.18115030472589635371802694520, −3.03482525132290781034712952445, −2.59905572369896572163032367723, −2.08967127068815525208415030472, −1.50809579182413042561356067714, 0, 0, 0, 0, 0, 1.50809579182413042561356067714, 2.08967127068815525208415030472, 2.59905572369896572163032367723, 3.03482525132290781034712952445, 3.18115030472589635371802694520, 4.04555556760300556346583373584, 4.63211455163064979737899825903, 4.68295572456568848726380961980, 5.11673631804307418603723291353, 5.29342614602785183353818983280, 6.22045722891093812723762630188, 6.30061561179429969377654018036, 6.93469958466916536339152157830, 7.38404150983480921923092180577, 7.77930575353127022717719805558, 8.444598562801082828116530343262, 8.493811020560654924164991290973, 8.804482240001268229157234513930, 8.968669119891794363502089843787, 9.113537941465847528066041821250, 9.758975601402302747241374336491, 10.11522513621733920078992331813, 10.16156449821809432477882806299, 10.45334061861574183526918956826, 11.12167393245733668684114048203