Dirichlet series
| L(s) = 1 | + 28·3-s + 200·5-s − 80·7-s − 112·9-s + 444·11-s − 1.35e3·13-s + 5.60e3·15-s − 3.40e3·17-s + 1.66e3·19-s − 2.24e3·21-s + 2.96e3·23-s + 2.25e4·25-s − 7.90e3·27-s − 8.27e3·29-s + 8.68e3·31-s + 1.24e4·33-s − 1.60e4·35-s + 1.13e4·37-s − 3.78e4·39-s + 280·41-s + 3.62e4·43-s − 2.24e4·45-s + 1.79e4·47-s − 2.58e4·49-s − 9.52e4·51-s + 2.89e4·53-s + 8.88e4·55-s + ⋯ |
| L(s) = 1 | + 1.79·3-s + 3.57·5-s − 0.617·7-s − 0.460·9-s + 1.10·11-s − 2.21·13-s + 6.42·15-s − 2.85·17-s + 1.06·19-s − 1.10·21-s + 1.16·23-s + 36/5·25-s − 2.08·27-s − 1.82·29-s + 1.62·31-s + 1.98·33-s − 2.20·35-s + 1.35·37-s − 3.98·39-s + 0.0260·41-s + 2.99·43-s − 1.64·45-s + 1.18·47-s − 1.53·49-s − 5.12·51-s + 1.41·53-s + 3.95·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 5^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.99171\times 10^{17}\) |
| Root analytic conductor: | \(12.9150\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 5^{8} \cdot 13^{8} ,\ ( \ : [5/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(353.2666637\) |
| \(L(\frac12)\) | \(\approx\) | \(353.2666637\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - p^{2} T )^{8} \) | |
| 13 | \( ( 1 + p^{2} T )^{8} \) | |
| good | 3 | \( 1 - 28 T + 896 T^{2} - 20324 T^{3} + 479540 T^{4} - 9048100 T^{5} + 59765632 p T^{6} - 328652444 p^{2} T^{7} + 603671974 p^{4} T^{8} - 328652444 p^{7} T^{9} + 59765632 p^{11} T^{10} - 9048100 p^{15} T^{11} + 479540 p^{20} T^{12} - 20324 p^{25} T^{13} + 896 p^{30} T^{14} - 28 p^{35} T^{15} + p^{40} T^{16} \) |
| 7 | \( 1 + 80 T + 4600 p T^{2} + 380976 p T^{3} + 482264348 T^{4} + 18986403472 T^{5} + 1539520544760 T^{6} - 194629475779952 T^{7} - 68075964702264826 T^{8} - 194629475779952 p^{5} T^{9} + 1539520544760 p^{10} T^{10} + 18986403472 p^{15} T^{11} + 482264348 p^{20} T^{12} + 380976 p^{26} T^{13} + 4600 p^{31} T^{14} + 80 p^{35} T^{15} + p^{40} T^{16} \) | |
| 11 | \( 1 - 444 T + 761224 T^{2} - 375958996 T^{3} + 289193525492 T^{4} - 133568093200276 T^{5} + 75493973403958872 T^{6} - 28787487260882689180 T^{7} + \)\(13\!\cdots\!94\)\( p T^{8} - 28787487260882689180 p^{5} T^{9} + 75493973403958872 p^{10} T^{10} - 133568093200276 p^{15} T^{11} + 289193525492 p^{20} T^{12} - 375958996 p^{25} T^{13} + 761224 p^{30} T^{14} - 444 p^{35} T^{15} + p^{40} T^{16} \) | |
| 17 | \( 1 + 200 p T + 10356744 T^{2} + 21180048088 T^{3} + 38988524762940 T^{4} + 59351274514391368 T^{5} + 83307927604904214456 T^{6} + \)\(10\!\cdots\!12\)\( T^{7} + \)\(13\!\cdots\!86\)\( T^{8} + \)\(10\!\cdots\!12\)\( p^{5} T^{9} + 83307927604904214456 p^{10} T^{10} + 59351274514391368 p^{15} T^{11} + 38988524762940 p^{20} T^{12} + 21180048088 p^{25} T^{13} + 10356744 p^{30} T^{14} + 200 p^{36} T^{15} + p^{40} T^{16} \) | |
| 19 | \( 1 - 1668 T + 9675000 T^{2} - 3481544396 T^{3} + 29742332114036 T^{4} + 20383593583720916 T^{5} + 96935909220236648872 T^{6} + \)\(36\!\cdots\!72\)\( T^{7} + \)\(34\!\cdots\!98\)\( T^{8} + \)\(36\!\cdots\!72\)\( p^{5} T^{9} + 96935909220236648872 p^{10} T^{10} + 20383593583720916 p^{15} T^{11} + 29742332114036 p^{20} T^{12} - 3481544396 p^{25} T^{13} + 9675000 p^{30} T^{14} - 1668 p^{35} T^{15} + p^{40} T^{16} \) | |
| 23 | \( 1 - 2964 T + 29741328 T^{2} - 69276494812 T^{3} + 464560793235156 T^{4} - 933247702664941228 T^{5} + \)\(48\!\cdots\!96\)\( T^{6} - \)\(84\!\cdots\!36\)\( T^{7} + \)\(36\!\cdots\!34\)\( T^{8} - \)\(84\!\cdots\!36\)\( p^{5} T^{9} + \)\(48\!\cdots\!96\)\( p^{10} T^{10} - 933247702664941228 p^{15} T^{11} + 464560793235156 p^{20} T^{12} - 69276494812 p^{25} T^{13} + 29741328 p^{30} T^{14} - 2964 p^{35} T^{15} + p^{40} T^{16} \) | |
| 29 | \( 1 + 8272 T + 103267496 T^{2} + 706339293808 T^{3} + 5691939004480508 T^{4} + 32203724814383417616 T^{5} + \)\(19\!\cdots\!80\)\( T^{6} + \)\(95\!\cdots\!72\)\( T^{7} + \)\(48\!\cdots\!26\)\( T^{8} + \)\(95\!\cdots\!72\)\( p^{5} T^{9} + \)\(19\!\cdots\!80\)\( p^{10} T^{10} + 32203724814383417616 p^{15} T^{11} + 5691939004480508 p^{20} T^{12} + 706339293808 p^{25} T^{13} + 103267496 p^{30} T^{14} + 8272 p^{35} T^{15} + p^{40} T^{16} \) | |
| 31 | \( 1 - 8684 T + 69505672 T^{2} - 109069532340 T^{3} + 2272901250962388 T^{4} - 15861591218495640100 T^{5} + \)\(16\!\cdots\!04\)\( T^{6} - \)\(34\!\cdots\!96\)\( T^{7} + \)\(24\!\cdots\!94\)\( T^{8} - \)\(34\!\cdots\!96\)\( p^{5} T^{9} + \)\(16\!\cdots\!04\)\( p^{10} T^{10} - 15861591218495640100 p^{15} T^{11} + 2272901250962388 p^{20} T^{12} - 109069532340 p^{25} T^{13} + 69505672 p^{30} T^{14} - 8684 p^{35} T^{15} + p^{40} T^{16} \) | |
| 37 | \( 1 - 11320 T + 251597960 T^{2} - 2439822754792 T^{3} + 36224665344517884 T^{4} - \)\(29\!\cdots\!40\)\( T^{5} + \)\(33\!\cdots\!16\)\( T^{6} - \)\(24\!\cdots\!80\)\( T^{7} + \)\(25\!\cdots\!58\)\( T^{8} - \)\(24\!\cdots\!80\)\( p^{5} T^{9} + \)\(33\!\cdots\!16\)\( p^{10} T^{10} - \)\(29\!\cdots\!40\)\( p^{15} T^{11} + 36224665344517884 p^{20} T^{12} - 2439822754792 p^{25} T^{13} + 251597960 p^{30} T^{14} - 11320 p^{35} T^{15} + p^{40} T^{16} \) | |
| 41 | \( 1 - 280 T + 353931432 T^{2} - 990688579336 T^{3} + 72578123104902204 T^{4} - \)\(39\!\cdots\!80\)\( T^{5} + \)\(11\!\cdots\!52\)\( T^{6} - \)\(64\!\cdots\!64\)\( T^{7} + \)\(15\!\cdots\!62\)\( T^{8} - \)\(64\!\cdots\!64\)\( p^{5} T^{9} + \)\(11\!\cdots\!52\)\( p^{10} T^{10} - \)\(39\!\cdots\!80\)\( p^{15} T^{11} + 72578123104902204 p^{20} T^{12} - 990688579336 p^{25} T^{13} + 353931432 p^{30} T^{14} - 280 p^{35} T^{15} + p^{40} T^{16} \) | |
| 43 | \( 1 - 36268 T + 1338192880 T^{2} - 28924618303732 T^{3} + 635821763191124084 T^{4} - \)\(10\!\cdots\!12\)\( T^{5} + \)\(17\!\cdots\!96\)\( T^{6} - \)\(22\!\cdots\!88\)\( T^{7} + \)\(30\!\cdots\!50\)\( T^{8} - \)\(22\!\cdots\!88\)\( p^{5} T^{9} + \)\(17\!\cdots\!96\)\( p^{10} T^{10} - \)\(10\!\cdots\!12\)\( p^{15} T^{11} + 635821763191124084 p^{20} T^{12} - 28924618303732 p^{25} T^{13} + 1338192880 p^{30} T^{14} - 36268 p^{35} T^{15} + p^{40} T^{16} \) | |
| 47 | \( 1 - 17936 T + 1053689352 T^{2} - 16193660829584 T^{3} + 454102464763308380 T^{4} - \)\(64\!\cdots\!64\)\( T^{5} + \)\(10\!\cdots\!52\)\( T^{6} - \)\(16\!\cdots\!44\)\( T^{7} + \)\(22\!\cdots\!14\)\( T^{8} - \)\(16\!\cdots\!44\)\( p^{5} T^{9} + \)\(10\!\cdots\!52\)\( p^{10} T^{10} - \)\(64\!\cdots\!64\)\( p^{15} T^{11} + 454102464763308380 p^{20} T^{12} - 16193660829584 p^{25} T^{13} + 1053689352 p^{30} T^{14} - 17936 p^{35} T^{15} + p^{40} T^{16} \) | |
| 53 | \( 1 - 28936 T + 1686528104 T^{2} - 10226920603000 T^{3} + 549012531352825564 T^{4} + \)\(14\!\cdots\!28\)\( T^{5} + \)\(12\!\cdots\!24\)\( T^{6} + \)\(75\!\cdots\!96\)\( T^{7} + \)\(11\!\cdots\!10\)\( T^{8} + \)\(75\!\cdots\!96\)\( p^{5} T^{9} + \)\(12\!\cdots\!24\)\( p^{10} T^{10} + \)\(14\!\cdots\!28\)\( p^{15} T^{11} + 549012531352825564 p^{20} T^{12} - 10226920603000 p^{25} T^{13} + 1686528104 p^{30} T^{14} - 28936 p^{35} T^{15} + p^{40} T^{16} \) | |
| 59 | \( 1 - 34668 T + 4397190104 T^{2} - 112921016869092 T^{3} + 8378813490641397940 T^{4} - \)\(16\!\cdots\!08\)\( T^{5} + \)\(96\!\cdots\!28\)\( T^{6} - \)\(15\!\cdots\!40\)\( T^{7} + \)\(79\!\cdots\!86\)\( T^{8} - \)\(15\!\cdots\!40\)\( p^{5} T^{9} + \)\(96\!\cdots\!28\)\( p^{10} T^{10} - \)\(16\!\cdots\!08\)\( p^{15} T^{11} + 8378813490641397940 p^{20} T^{12} - 112921016869092 p^{25} T^{13} + 4397190104 p^{30} T^{14} - 34668 p^{35} T^{15} + p^{40} T^{16} \) | |
| 61 | \( 1 - 62976 T + 6038241512 T^{2} - 314186062082816 T^{3} + 16648308519496474748 T^{4} - \)\(70\!\cdots\!92\)\( T^{5} + \)\(27\!\cdots\!84\)\( T^{6} - \)\(92\!\cdots\!08\)\( T^{7} + \)\(28\!\cdots\!94\)\( T^{8} - \)\(92\!\cdots\!08\)\( p^{5} T^{9} + \)\(27\!\cdots\!84\)\( p^{10} T^{10} - \)\(70\!\cdots\!92\)\( p^{15} T^{11} + 16648308519496474748 p^{20} T^{12} - 314186062082816 p^{25} T^{13} + 6038241512 p^{30} T^{14} - 62976 p^{35} T^{15} + p^{40} T^{16} \) | |
| 67 | \( 1 - 181688 T + 20395512408 T^{2} - 1634678892642552 T^{3} + 1581546169835725300 p T^{4} - \)\(57\!\cdots\!04\)\( T^{5} + \)\(27\!\cdots\!40\)\( T^{6} - \)\(11\!\cdots\!16\)\( T^{7} + \)\(44\!\cdots\!58\)\( T^{8} - \)\(11\!\cdots\!16\)\( p^{5} T^{9} + \)\(27\!\cdots\!40\)\( p^{10} T^{10} - \)\(57\!\cdots\!04\)\( p^{15} T^{11} + 1581546169835725300 p^{21} T^{12} - 1634678892642552 p^{25} T^{13} + 20395512408 p^{30} T^{14} - 181688 p^{35} T^{15} + p^{40} T^{16} \) | |
| 71 | \( 1 - 6036 T + 2769112648 T^{2} + 43187104999636 T^{3} + 4275280666880342292 T^{4} + \)\(11\!\cdots\!60\)\( T^{5} + \)\(32\!\cdots\!44\)\( T^{6} + \)\(66\!\cdots\!80\)\( T^{7} + \)\(88\!\cdots\!70\)\( T^{8} + \)\(66\!\cdots\!80\)\( p^{5} T^{9} + \)\(32\!\cdots\!44\)\( p^{10} T^{10} + \)\(11\!\cdots\!60\)\( p^{15} T^{11} + 4275280666880342292 p^{20} T^{12} + 43187104999636 p^{25} T^{13} + 2769112648 p^{30} T^{14} - 6036 p^{35} T^{15} + p^{40} T^{16} \) | |
| 73 | \( 1 - 23000 T + 11079038568 T^{2} - 209753507202536 T^{3} + 58633041394100479260 T^{4} - \)\(90\!\cdots\!84\)\( T^{5} + \)\(19\!\cdots\!76\)\( T^{6} - \)\(25\!\cdots\!84\)\( T^{7} + \)\(47\!\cdots\!30\)\( T^{8} - \)\(25\!\cdots\!84\)\( p^{5} T^{9} + \)\(19\!\cdots\!76\)\( p^{10} T^{10} - \)\(90\!\cdots\!84\)\( p^{15} T^{11} + 58633041394100479260 p^{20} T^{12} - 209753507202536 p^{25} T^{13} + 11079038568 p^{30} T^{14} - 23000 p^{35} T^{15} + p^{40} T^{16} \) | |
| 79 | \( 1 - 24392 T + 11976772760 T^{2} - 176200596709320 T^{3} + 80825149975898804828 T^{4} - \)\(10\!\cdots\!68\)\( T^{5} + \)\(38\!\cdots\!56\)\( T^{6} - \)\(43\!\cdots\!16\)\( T^{7} + \)\(13\!\cdots\!78\)\( T^{8} - \)\(43\!\cdots\!16\)\( p^{5} T^{9} + \)\(38\!\cdots\!56\)\( p^{10} T^{10} - \)\(10\!\cdots\!68\)\( p^{15} T^{11} + 80825149975898804828 p^{20} T^{12} - 176200596709320 p^{25} T^{13} + 11976772760 p^{30} T^{14} - 24392 p^{35} T^{15} + p^{40} T^{16} \) | |
| 83 | \( 1 + 34416 T + 11304483112 T^{2} + 474336569074608 T^{3} + 90859302222231330620 T^{4} + \)\(42\!\cdots\!60\)\( T^{5} + \)\(50\!\cdots\!60\)\( T^{6} + \)\(23\!\cdots\!52\)\( T^{7} + \)\(22\!\cdots\!90\)\( T^{8} + \)\(23\!\cdots\!52\)\( p^{5} T^{9} + \)\(50\!\cdots\!60\)\( p^{10} T^{10} + \)\(42\!\cdots\!60\)\( p^{15} T^{11} + 90859302222231330620 p^{20} T^{12} + 474336569074608 p^{25} T^{13} + 11304483112 p^{30} T^{14} + 34416 p^{35} T^{15} + p^{40} T^{16} \) | |
| 89 | \( 1 - 57328 T + 19138368568 T^{2} - 972850592832848 T^{3} + \)\(21\!\cdots\!52\)\( T^{4} - \)\(93\!\cdots\!68\)\( T^{5} + \)\(16\!\cdots\!64\)\( T^{6} - \)\(62\!\cdots\!60\)\( T^{7} + \)\(99\!\cdots\!30\)\( T^{8} - \)\(62\!\cdots\!60\)\( p^{5} T^{9} + \)\(16\!\cdots\!64\)\( p^{10} T^{10} - \)\(93\!\cdots\!68\)\( p^{15} T^{11} + \)\(21\!\cdots\!52\)\( p^{20} T^{12} - 972850592832848 p^{25} T^{13} + 19138368568 p^{30} T^{14} - 57328 p^{35} T^{15} + p^{40} T^{16} \) | |
| 97 | \( 1 + 210176 T + 67804956568 T^{2} + 10811134130154880 T^{3} + \)\(20\!\cdots\!12\)\( T^{4} + \)\(25\!\cdots\!20\)\( T^{5} + \)\(34\!\cdots\!88\)\( T^{6} + \)\(34\!\cdots\!32\)\( T^{7} + \)\(36\!\cdots\!22\)\( T^{8} + \)\(34\!\cdots\!32\)\( p^{5} T^{9} + \)\(34\!\cdots\!88\)\( p^{10} T^{10} + \)\(25\!\cdots\!20\)\( p^{15} T^{11} + \)\(20\!\cdots\!12\)\( p^{20} T^{12} + 10811134130154880 p^{25} T^{13} + 67804956568 p^{30} T^{14} + 210176 p^{35} T^{15} + p^{40} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.56454275311475411696528597675, −2.95803095264685678720963498476, −2.91052999863499104003929454083, −2.90923027700823583707085432288, −2.87516377543227984559618668717, −2.79233597047458993475087619146, −2.72227706519368517948383053177, −2.68968565477771899270993273119, −2.46744236962445873607636243544, −2.15060172628228178972563519131, −2.14898933595387318611623111745, −2.14110534286536399537533669131, −2.00504158905218732812724840976, −1.84899161875071574005733480558, −1.74686199877151986589039446725, −1.54944242424976798151220852219, −1.53015521709451205255651743328, −1.06002313213671700990316318825, −0.905032106887749859220070793891, −0.869106633599175039779306379344, −0.59658745386625113544534607724, −0.58421779125272143976859608299, −0.53675286964687923922203922541, −0.49370225499344122617206164253, −0.29310974343471036626174634490, 0.29310974343471036626174634490, 0.49370225499344122617206164253, 0.53675286964687923922203922541, 0.58421779125272143976859608299, 0.59658745386625113544534607724, 0.869106633599175039779306379344, 0.905032106887749859220070793891, 1.06002313213671700990316318825, 1.53015521709451205255651743328, 1.54944242424976798151220852219, 1.74686199877151986589039446725, 1.84899161875071574005733480558, 2.00504158905218732812724840976, 2.14110534286536399537533669131, 2.14898933595387318611623111745, 2.15060172628228178972563519131, 2.46744236962445873607636243544, 2.68968565477771899270993273119, 2.72227706519368517948383053177, 2.79233597047458993475087619146, 2.87516377543227984559618668717, 2.90923027700823583707085432288, 2.91052999863499104003929454083, 2.95803095264685678720963498476, 3.56454275311475411696528597675
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.