Dirichlet series
| L(s) = 1 | − 16·2-s − 24·3-s + 144·4-s + 5-s + 384·6-s + 22·7-s − 960·8-s + 324·9-s − 16·10-s − 77·11-s − 3.45e3·12-s + 29·13-s − 352·14-s − 24·15-s + 5.28e3·16-s + 59·17-s − 5.18e3·18-s − 150·19-s + 144·20-s − 528·21-s + 1.23e3·22-s − 269·23-s + 2.30e4·24-s − 363·25-s − 464·26-s − 3.24e3·27-s + 3.16e3·28-s + ⋯ |
| L(s) = 1 | − 5.65·2-s − 4.61·3-s + 18·4-s + 0.0894·5-s + 26.1·6-s + 1.18·7-s − 42.4·8-s + 12·9-s − 0.505·10-s − 2.11·11-s − 83.1·12-s + 0.618·13-s − 6.71·14-s − 0.413·15-s + 82.5·16-s + 0.841·17-s − 67.8·18-s − 1.81·19-s + 1.60·20-s − 5.48·21-s + 11.9·22-s − 2.43·23-s + 195.·24-s − 2.90·25-s − 3.49·26-s − 23.0·27-s + 21.3·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 131^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 131^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 131^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.13948\times 10^{13}\) |
| Root analytic conductor: | \(6.80995\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 131^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T )^{8} \) |
| 3 | \( ( 1 + p T )^{8} \) | |
| 131 | \( ( 1 - p T )^{8} \) | |
| good | 5 | \( 1 - T + 364 T^{2} + 1096 T^{3} + 80564 T^{4} + 335757 T^{5} + 14859768 T^{6} + 53703072 T^{7} + 2128020254 T^{8} + 53703072 p^{3} T^{9} + 14859768 p^{6} T^{10} + 335757 p^{9} T^{11} + 80564 p^{12} T^{12} + 1096 p^{15} T^{13} + 364 p^{18} T^{14} - p^{21} T^{15} + p^{24} T^{16} \) |
| 7 | \( 1 - 22 T + 968 T^{2} - 16252 T^{3} + 80006 p T^{4} - 10681932 T^{5} + 283222833 T^{6} - 4311364726 T^{7} + 98531385728 T^{8} - 4311364726 p^{3} T^{9} + 283222833 p^{6} T^{10} - 10681932 p^{9} T^{11} + 80006 p^{13} T^{12} - 16252 p^{15} T^{13} + 968 p^{18} T^{14} - 22 p^{21} T^{15} + p^{24} T^{16} \) | |
| 11 | \( 1 + 7 p T + 9477 T^{2} + 536651 T^{3} + 38783600 T^{4} + 1777846207 T^{5} + 95003513025 T^{6} + 3602924358438 T^{7} + 153607825684608 T^{8} + 3602924358438 p^{3} T^{9} + 95003513025 p^{6} T^{10} + 1777846207 p^{9} T^{11} + 38783600 p^{12} T^{12} + 536651 p^{15} T^{13} + 9477 p^{18} T^{14} + 7 p^{22} T^{15} + p^{24} T^{16} \) | |
| 13 | \( 1 - 29 T + 10323 T^{2} - 294326 T^{3} + 51143113 T^{4} - 113878758 p T^{5} + 171187910228 T^{6} - 4803241940047 T^{7} + 430815938247918 T^{8} - 4803241940047 p^{3} T^{9} + 171187910228 p^{6} T^{10} - 113878758 p^{10} T^{11} + 51143113 p^{12} T^{12} - 294326 p^{15} T^{13} + 10323 p^{18} T^{14} - 29 p^{21} T^{15} + p^{24} T^{16} \) | |
| 17 | \( 1 - 59 T + 17863 T^{2} - 972673 T^{3} + 183450758 T^{4} - 8889403791 T^{5} + 1334832979329 T^{6} - 57875658368016 T^{7} + 7334431399058588 T^{8} - 57875658368016 p^{3} T^{9} + 1334832979329 p^{6} T^{10} - 8889403791 p^{9} T^{11} + 183450758 p^{12} T^{12} - 972673 p^{15} T^{13} + 17863 p^{18} T^{14} - 59 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 + 150 T + 37578 T^{2} + 5065262 T^{3} + 747780816 T^{4} + 81837218663 T^{5} + 9284133783973 T^{6} + 835370328157446 T^{7} + 76875370308931924 T^{8} + 835370328157446 p^{3} T^{9} + 9284133783973 p^{6} T^{10} + 81837218663 p^{9} T^{11} + 747780816 p^{12} T^{12} + 5065262 p^{15} T^{13} + 37578 p^{18} T^{14} + 150 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( 1 + 269 T + 95925 T^{2} + 19545550 T^{3} + 3994972377 T^{4} + 645941375991 T^{5} + 95557849958617 T^{6} + 12480328417128466 T^{7} + 1444358662297626816 T^{8} + 12480328417128466 p^{3} T^{9} + 95557849958617 p^{6} T^{10} + 645941375991 p^{9} T^{11} + 3994972377 p^{12} T^{12} + 19545550 p^{15} T^{13} + 95925 p^{18} T^{14} + 269 p^{21} T^{15} + p^{24} T^{16} \) | |
| 29 | \( 1 + 123 T + 66432 T^{2} - 1723899 T^{3} + 1225932164 T^{4} - 280905542211 T^{5} + 43701377191515 T^{6} - 4740602308189668 T^{7} + 1753683997577997666 T^{8} - 4740602308189668 p^{3} T^{9} + 43701377191515 p^{6} T^{10} - 280905542211 p^{9} T^{11} + 1225932164 p^{12} T^{12} - 1723899 p^{15} T^{13} + 66432 p^{18} T^{14} + 123 p^{21} T^{15} + p^{24} T^{16} \) | |
| 31 | \( 1 + 86 T + 100155 T^{2} + 2172843 T^{3} + 5050406421 T^{4} - 28435025010 T^{5} + 211989271102988 T^{6} + 687187761302269 T^{7} + 7433607217841637234 T^{8} + 687187761302269 p^{3} T^{9} + 211989271102988 p^{6} T^{10} - 28435025010 p^{9} T^{11} + 5050406421 p^{12} T^{12} + 2172843 p^{15} T^{13} + 100155 p^{18} T^{14} + 86 p^{21} T^{15} + p^{24} T^{16} \) | |
| 37 | \( 1 - 412 T + 306683 T^{2} - 88264079 T^{3} + 40715231537 T^{4} - 9421585418792 T^{5} + 3387376410014298 T^{6} - 658276497719611521 T^{7} + \)\(19\!\cdots\!26\)\( T^{8} - 658276497719611521 p^{3} T^{9} + 3387376410014298 p^{6} T^{10} - 9421585418792 p^{9} T^{11} + 40715231537 p^{12} T^{12} - 88264079 p^{15} T^{13} + 306683 p^{18} T^{14} - 412 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( 1 + 114 T + 382011 T^{2} + 27027990 T^{3} + 70377446957 T^{4} + 3412249207095 T^{5} + 8297043339404625 T^{6} + 305953258555130637 T^{7} + \)\(67\!\cdots\!00\)\( T^{8} + 305953258555130637 p^{3} T^{9} + 8297043339404625 p^{6} T^{10} + 3412249207095 p^{9} T^{11} + 70377446957 p^{12} T^{12} + 27027990 p^{15} T^{13} + 382011 p^{18} T^{14} + 114 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( 1 - 1087 T + 731934 T^{2} - 358724021 T^{3} + 151543979973 T^{4} - 56724501624905 T^{5} + 19918483419272210 T^{6} - 6427094485124863059 T^{7} + \)\(19\!\cdots\!40\)\( T^{8} - 6427094485124863059 p^{3} T^{9} + 19918483419272210 p^{6} T^{10} - 56724501624905 p^{9} T^{11} + 151543979973 p^{12} T^{12} - 358724021 p^{15} T^{13} + 731934 p^{18} T^{14} - 1087 p^{21} T^{15} + p^{24} T^{16} \) | |
| 47 | \( 1 + 442 T + 377167 T^{2} + 122024178 T^{3} + 58371359487 T^{4} + 273499758501 p T^{5} + 5495073470894769 T^{6} + 885002223254213795 T^{7} + \)\(49\!\cdots\!40\)\( T^{8} + 885002223254213795 p^{3} T^{9} + 5495073470894769 p^{6} T^{10} + 273499758501 p^{10} T^{11} + 58371359487 p^{12} T^{12} + 122024178 p^{15} T^{13} + 377167 p^{18} T^{14} + 442 p^{21} T^{15} + p^{24} T^{16} \) | |
| 53 | \( 1 - 5 T + 1043312 T^{2} + 6909324 T^{3} + 493456304166 T^{4} + 6231421237657 T^{5} + 138488911438808406 T^{6} + 1906154680977360596 T^{7} + \)\(25\!\cdots\!38\)\( T^{8} + 1906154680977360596 p^{3} T^{9} + 138488911438808406 p^{6} T^{10} + 6231421237657 p^{9} T^{11} + 493456304166 p^{12} T^{12} + 6909324 p^{15} T^{13} + 1043312 p^{18} T^{14} - 5 p^{21} T^{15} + p^{24} T^{16} \) | |
| 59 | \( 1 - 252 T + 926166 T^{2} - 233643172 T^{3} + 456641826554 T^{4} - 110100678101321 T^{5} + 152074554817721089 T^{6} - 32954380540272216028 T^{7} + \)\(36\!\cdots\!20\)\( T^{8} - 32954380540272216028 p^{3} T^{9} + 152074554817721089 p^{6} T^{10} - 110100678101321 p^{9} T^{11} + 456641826554 p^{12} T^{12} - 233643172 p^{15} T^{13} + 926166 p^{18} T^{14} - 252 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 - 1482 T + 2168477 T^{2} - 2046974027 T^{3} + 1789979071129 T^{4} - 1259429166046072 T^{5} + 814706778642041868 T^{6} - \)\(45\!\cdots\!01\)\( T^{7} + \)\(23\!\cdots\!34\)\( T^{8} - \)\(45\!\cdots\!01\)\( p^{3} T^{9} + 814706778642041868 p^{6} T^{10} - 1259429166046072 p^{9} T^{11} + 1789979071129 p^{12} T^{12} - 2046974027 p^{15} T^{13} + 2168477 p^{18} T^{14} - 1482 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( 1 - 330 T + 1389173 T^{2} - 73442227 T^{3} + 746211826411 T^{4} + 220339931237950 T^{5} + 209316465191992716 T^{6} + \)\(17\!\cdots\!15\)\( T^{7} + \)\(49\!\cdots\!14\)\( T^{8} + \)\(17\!\cdots\!15\)\( p^{3} T^{9} + 209316465191992716 p^{6} T^{10} + 220339931237950 p^{9} T^{11} + 746211826411 p^{12} T^{12} - 73442227 p^{15} T^{13} + 1389173 p^{18} T^{14} - 330 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( 1 + 2946 T + 4958066 T^{2} + 5421540848 T^{3} + 4107810024024 T^{4} + 1897451051211639 T^{5} + 147327808428953264 T^{6} - \)\(59\!\cdots\!35\)\( T^{7} - \)\(53\!\cdots\!26\)\( T^{8} - \)\(59\!\cdots\!35\)\( p^{3} T^{9} + 147327808428953264 p^{6} T^{10} + 1897451051211639 p^{9} T^{11} + 4107810024024 p^{12} T^{12} + 5421540848 p^{15} T^{13} + 4958066 p^{18} T^{14} + 2946 p^{21} T^{15} + p^{24} T^{16} \) | |
| 73 | \( 1 + 214 T + 803543 T^{2} + 344902388 T^{3} + 375316779363 T^{4} + 64752623153620 T^{5} + 121273780111451621 T^{6} - 10028668506080112366 T^{7} + \)\(25\!\cdots\!40\)\( T^{8} - 10028668506080112366 p^{3} T^{9} + 121273780111451621 p^{6} T^{10} + 64752623153620 p^{9} T^{11} + 375316779363 p^{12} T^{12} + 344902388 p^{15} T^{13} + 803543 p^{18} T^{14} + 214 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( 1 + 64 T + 806187 T^{2} - 326189714 T^{3} - 13343939087 T^{4} - 215688457850142 T^{5} - 73384113948806011 T^{6} + 90741764640821364068 T^{7} + \)\(15\!\cdots\!20\)\( T^{8} + 90741764640821364068 p^{3} T^{9} - 73384113948806011 p^{6} T^{10} - 215688457850142 p^{9} T^{11} - 13343939087 p^{12} T^{12} - 326189714 p^{15} T^{13} + 806187 p^{18} T^{14} + 64 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( 1 + 276 T + 443010 T^{2} - 21915478 T^{3} + 378044841434 T^{4} + 309171417073357 T^{5} + 433597204776358432 T^{6} + 67787411801515493027 T^{7} + \)\(13\!\cdots\!66\)\( T^{8} + 67787411801515493027 p^{3} T^{9} + 433597204776358432 p^{6} T^{10} + 309171417073357 p^{9} T^{11} + 378044841434 p^{12} T^{12} - 21915478 p^{15} T^{13} + 443010 p^{18} T^{14} + 276 p^{21} T^{15} + p^{24} T^{16} \) | |
| 89 | \( 1 + 3177 T + 7900144 T^{2} + 13711980228 T^{3} + 20564901101726 T^{4} + 25498966253162223 T^{5} + 28468588470408203394 T^{6} + \)\(27\!\cdots\!72\)\( T^{7} + \)\(24\!\cdots\!50\)\( T^{8} + \)\(27\!\cdots\!72\)\( p^{3} T^{9} + 28468588470408203394 p^{6} T^{10} + 25498966253162223 p^{9} T^{11} + 20564901101726 p^{12} T^{12} + 13711980228 p^{15} T^{13} + 7900144 p^{18} T^{14} + 3177 p^{21} T^{15} + p^{24} T^{16} \) | |
| 97 | \( 1 - 200 T + 858070 T^{2} + 363324096 T^{3} + 1572946895813 T^{4} + 826719650275552 T^{5} + 1311900449939618778 T^{6} + \)\(69\!\cdots\!28\)\( T^{7} + \)\(19\!\cdots\!00\)\( T^{8} + \)\(69\!\cdots\!28\)\( p^{3} T^{9} + 1311900449939618778 p^{6} T^{10} + 826719650275552 p^{9} T^{11} + 1572946895813 p^{12} T^{12} + 363324096 p^{15} T^{13} + 858070 p^{18} T^{14} - 200 p^{21} T^{15} + p^{24} T^{16} \) | |
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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
−4.67262968470950530728718625399, −4.17134553387290284962840041334, −4.15839524801055639299212167926, −4.09377679188118446059486530872, −4.01070991930256321784583025580, −3.98059217051590044397314733032, −3.79773994986688104433043088933, −3.70094810040386762727323980457, −3.69307577664508700584613002713, −2.98199816339894508193987004952, −2.71659706104812186623916428420, −2.68153526371653568462055910254, −2.51821699423883497461446994450, −2.50124074654513069855393180084, −2.49074239361508863580163376416, −2.25140305467821717106393802496, −2.15590225407040926310686033921, −1.66055682094269599919632330173, −1.52185960618223644522113264644, −1.37388572967766001788558170868, −1.31509495220732821936489013283, −1.28471700998143382109865750817, −1.25784190548049875563985383493, −1.05969446259124148925892021013, −0.947592548915831594772698418071, 0, 0, 0, 0, 0, 0, 0, 0, 0.947592548915831594772698418071, 1.05969446259124148925892021013, 1.25784190548049875563985383493, 1.28471700998143382109865750817, 1.31509495220732821936489013283, 1.37388572967766001788558170868, 1.52185960618223644522113264644, 1.66055682094269599919632330173, 2.15590225407040926310686033921, 2.25140305467821717106393802496, 2.49074239361508863580163376416, 2.50124074654513069855393180084, 2.51821699423883497461446994450, 2.68153526371653568462055910254, 2.71659706104812186623916428420, 2.98199816339894508193987004952, 3.69307577664508700584613002713, 3.70094810040386762727323980457, 3.79773994986688104433043088933, 3.98059217051590044397314733032, 4.01070991930256321784583025580, 4.09377679188118446059486530872, 4.15839524801055639299212167926, 4.17134553387290284962840041334, 4.67262968470950530728718625399
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.