Dirichlet series
| L(s) = 1 | + 8·2-s − 27·3-s + 12·4-s + 41·5-s − 216·6-s + 7-s − 59·8-s + 405·9-s + 328·10-s + 37·11-s − 324·12-s + 8·14-s − 1.10e3·15-s − 200·16-s − 134·17-s + 3.24e3·18-s − 72·19-s + 492·20-s − 27·21-s + 296·22-s + 226·23-s + 1.59e3·24-s + 584·25-s − 4.45e3·27-s + 12·28-s − 547·29-s − 8.85e3·30-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 5.19·3-s + 3/2·4-s + 3.66·5-s − 14.6·6-s + 0.0539·7-s − 2.60·8-s + 15·9-s + 10.3·10-s + 1.01·11-s − 7.79·12-s + 0.152·14-s − 19.0·15-s − 3.12·16-s − 1.91·17-s + 42.4·18-s − 0.869·19-s + 5.50·20-s − 0.280·21-s + 2.86·22-s + 2.04·23-s + 13.5·24-s + 4.67·25-s − 31.7·27-s + 0.0809·28-s − 3.50·29-s − 53.8·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 13^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 13^{18}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(3^{9} \cdot 13^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.91807\times 10^{13}\) |
| Root analytic conductor: | \(5.46936\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 3^{9} \cdot 13^{18} ,\ ( \ : [3/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(36.32284332\) |
| \(L(\frac12)\) | \(\approx\) | \(36.32284332\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p T )^{9} \) |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - p^{3} T + 13 p^{2} T^{2} - 261 T^{3} + 149 p^{3} T^{4} - 4809 T^{5} + 17751 T^{6} - 30071 p T^{7} + 11903 p^{4} T^{8} - 69739 p^{3} T^{9} + 11903 p^{7} T^{10} - 30071 p^{7} T^{11} + 17751 p^{9} T^{12} - 4809 p^{12} T^{13} + 149 p^{18} T^{14} - 261 p^{18} T^{15} + 13 p^{23} T^{16} - p^{27} T^{17} + p^{27} T^{18} \) |
| 5 | \( 1 - 41 T + 1097 T^{2} - 22813 T^{3} + 407739 T^{4} - 6387089 T^{5} + 92503276 T^{6} - 1243056011 T^{7} + 15569256043 T^{8} - 180245321811 T^{9} + 15569256043 p^{3} T^{10} - 1243056011 p^{6} T^{11} + 92503276 p^{9} T^{12} - 6387089 p^{12} T^{13} + 407739 p^{15} T^{14} - 22813 p^{18} T^{15} + 1097 p^{21} T^{16} - 41 p^{24} T^{17} + p^{27} T^{18} \) | |
| 7 | \( 1 - T + 1223 T^{2} - 2029 T^{3} + 123315 p T^{4} - 106831 p T^{5} + 66703926 p T^{6} - 3091881 p^{2} T^{7} + 27857015069 p T^{8} - 9167889093 p T^{9} + 27857015069 p^{4} T^{10} - 3091881 p^{8} T^{11} + 66703926 p^{10} T^{12} - 106831 p^{13} T^{13} + 123315 p^{16} T^{14} - 2029 p^{18} T^{15} + 1223 p^{21} T^{16} - p^{24} T^{17} + p^{27} T^{18} \) | |
| 11 | \( 1 - 37 T + 5344 T^{2} - 236518 T^{3} + 17901017 T^{4} - 63735945 p T^{5} + 42666463784 T^{6} - 1453902639631 T^{7} + 73009122696568 T^{8} - 2268390689145787 T^{9} + 73009122696568 p^{3} T^{10} - 1453902639631 p^{6} T^{11} + 42666463784 p^{9} T^{12} - 63735945 p^{13} T^{13} + 17901017 p^{15} T^{14} - 236518 p^{18} T^{15} + 5344 p^{21} T^{16} - 37 p^{24} T^{17} + p^{27} T^{18} \) | |
| 17 | \( 1 + 134 T + 31870 T^{2} + 2895632 T^{3} + 389871107 T^{4} + 25426400680 T^{5} + 2595913829472 T^{6} + 127839526448684 T^{7} + 12429944904049430 T^{8} + 559326357383412308 T^{9} + 12429944904049430 p^{3} T^{10} + 127839526448684 p^{6} T^{11} + 2595913829472 p^{9} T^{12} + 25426400680 p^{12} T^{13} + 389871107 p^{15} T^{14} + 2895632 p^{18} T^{15} + 31870 p^{21} T^{16} + 134 p^{24} T^{17} + p^{27} T^{18} \) | |
| 19 | \( 1 + 72 T + 27564 T^{2} + 1147684 T^{3} + 20554227 p T^{4} + 9456244878 T^{5} + 3864610586734 T^{6} + 39462617496456 T^{7} + 1598285766902820 p T^{8} + 156420251845519228 T^{9} + 1598285766902820 p^{4} T^{10} + 39462617496456 p^{6} T^{11} + 3864610586734 p^{9} T^{12} + 9456244878 p^{12} T^{13} + 20554227 p^{16} T^{14} + 1147684 p^{18} T^{15} + 27564 p^{21} T^{16} + 72 p^{24} T^{17} + p^{27} T^{18} \) | |
| 23 | \( 1 - 226 T + 81815 T^{2} - 16258082 T^{3} + 3468064857 T^{4} - 557247960198 T^{5} + 90899581520090 T^{6} - 12044943239754990 T^{7} + 1588739302979220263 T^{8} - \)\(17\!\cdots\!68\)\( T^{9} + 1588739302979220263 p^{3} T^{10} - 12044943239754990 p^{6} T^{11} + 90899581520090 p^{9} T^{12} - 557247960198 p^{12} T^{13} + 3468064857 p^{15} T^{14} - 16258082 p^{18} T^{15} + 81815 p^{21} T^{16} - 226 p^{24} T^{17} + p^{27} T^{18} \) | |
| 29 | \( 1 + 547 T + 257397 T^{2} + 76666599 T^{3} + 20293099631 T^{4} + 4094154725297 T^{5} + 763499956425928 T^{6} + 117635069996325623 T^{7} + 18495766143723554001 T^{8} + \)\(27\!\cdots\!45\)\( T^{9} + 18495766143723554001 p^{3} T^{10} + 117635069996325623 p^{6} T^{11} + 763499956425928 p^{9} T^{12} + 4094154725297 p^{12} T^{13} + 20293099631 p^{15} T^{14} + 76666599 p^{18} T^{15} + 257397 p^{21} T^{16} + 547 p^{24} T^{17} + p^{27} T^{18} \) | |
| 31 | \( 1 + 521 T + 293139 T^{2} + 96841141 T^{3} + 30937308663 T^{4} + 241913814859 p T^{5} + 1735634762957196 T^{6} + 340137934421353947 T^{7} + 64997937679040291581 T^{8} + \)\(11\!\cdots\!05\)\( T^{9} + 64997937679040291581 p^{3} T^{10} + 340137934421353947 p^{6} T^{11} + 1735634762957196 p^{9} T^{12} + 241913814859 p^{13} T^{13} + 30937308663 p^{15} T^{14} + 96841141 p^{18} T^{15} + 293139 p^{21} T^{16} + 521 p^{24} T^{17} + p^{27} T^{18} \) | |
| 37 | \( 1 - 584 T + 402607 T^{2} - 171417710 T^{3} + 70390531675 T^{4} - 23711716261954 T^{5} + 7321264414358712 T^{6} - 2049337722253604358 T^{7} + \)\(51\!\cdots\!49\)\( T^{8} - \)\(12\!\cdots\!04\)\( T^{9} + \)\(51\!\cdots\!49\)\( p^{3} T^{10} - 2049337722253604358 p^{6} T^{11} + 7321264414358712 p^{9} T^{12} - 23711716261954 p^{12} T^{13} + 70390531675 p^{15} T^{14} - 171417710 p^{18} T^{15} + 402607 p^{21} T^{16} - 584 p^{24} T^{17} + p^{27} T^{18} \) | |
| 41 | \( 1 - 482 T + 419110 T^{2} - 170189112 T^{3} + 88736153509 T^{4} - 29827436903094 T^{5} + 11832109071533322 T^{6} - 3396334856238665746 T^{7} + \)\(11\!\cdots\!98\)\( T^{8} - \)\(27\!\cdots\!56\)\( T^{9} + \)\(11\!\cdots\!98\)\( p^{3} T^{10} - 3396334856238665746 p^{6} T^{11} + 11832109071533322 p^{9} T^{12} - 29827436903094 p^{12} T^{13} + 88736153509 p^{15} T^{14} - 170189112 p^{18} T^{15} + 419110 p^{21} T^{16} - 482 p^{24} T^{17} + p^{27} T^{18} \) | |
| 43 | \( 1 - 158 T + 414734 T^{2} - 70335996 T^{3} + 91129375254 T^{4} - 14698619715102 T^{5} + 13437400325294869 T^{6} - 1968909046011777708 T^{7} + \)\(14\!\cdots\!28\)\( T^{8} - \)\(18\!\cdots\!08\)\( T^{9} + \)\(14\!\cdots\!28\)\( p^{3} T^{10} - 1968909046011777708 p^{6} T^{11} + 13437400325294869 p^{9} T^{12} - 14698619715102 p^{12} T^{13} + 91129375254 p^{15} T^{14} - 70335996 p^{18} T^{15} + 414734 p^{21} T^{16} - 158 p^{24} T^{17} + p^{27} T^{18} \) | |
| 47 | \( 1 - 1500 T + 1371544 T^{2} - 916634374 T^{3} + 489838679415 T^{4} - 220656283954330 T^{5} + 87041732536135146 T^{6} - 31223835492605943510 T^{7} + \)\(10\!\cdots\!30\)\( T^{8} - \)\(34\!\cdots\!08\)\( T^{9} + \)\(10\!\cdots\!30\)\( p^{3} T^{10} - 31223835492605943510 p^{6} T^{11} + 87041732536135146 p^{9} T^{12} - 220656283954330 p^{12} T^{13} + 489838679415 p^{15} T^{14} - 916634374 p^{18} T^{15} + 1371544 p^{21} T^{16} - 1500 p^{24} T^{17} + p^{27} T^{18} \) | |
| 53 | \( 1 - 1399 T + 1838846 T^{2} - 1568203054 T^{3} + 1236563700767 T^{4} - 774676352395257 T^{5} + 451502726866177032 T^{6} - \)\(22\!\cdots\!09\)\( T^{7} + \)\(10\!\cdots\!84\)\( T^{8} - \)\(40\!\cdots\!57\)\( T^{9} + \)\(10\!\cdots\!84\)\( p^{3} T^{10} - \)\(22\!\cdots\!09\)\( p^{6} T^{11} + 451502726866177032 p^{9} T^{12} - 774676352395257 p^{12} T^{13} + 1236563700767 p^{15} T^{14} - 1568203054 p^{18} T^{15} + 1838846 p^{21} T^{16} - 1399 p^{24} T^{17} + p^{27} T^{18} \) | |
| 59 | \( 1 - 1541 T + 1980051 T^{2} - 1607792373 T^{3} + 1229833622732 T^{4} - 735479443332906 T^{5} + 441803036901505636 T^{6} - \)\(22\!\cdots\!55\)\( T^{7} + \)\(11\!\cdots\!02\)\( T^{8} - \)\(50\!\cdots\!78\)\( T^{9} + \)\(11\!\cdots\!02\)\( p^{3} T^{10} - \)\(22\!\cdots\!55\)\( p^{6} T^{11} + 441803036901505636 p^{9} T^{12} - 735479443332906 p^{12} T^{13} + 1229833622732 p^{15} T^{14} - 1607792373 p^{18} T^{15} + 1980051 p^{21} T^{16} - 1541 p^{24} T^{17} + p^{27} T^{18} \) | |
| 61 | \( 1 - 2092 T + 2899914 T^{2} - 2746264024 T^{3} + 2107526963381 T^{4} - 1297704122709412 T^{5} + 700467654937121908 T^{6} - \)\(33\!\cdots\!26\)\( T^{7} + \)\(15\!\cdots\!76\)\( T^{8} - \)\(69\!\cdots\!56\)\( T^{9} + \)\(15\!\cdots\!76\)\( p^{3} T^{10} - \)\(33\!\cdots\!26\)\( p^{6} T^{11} + 700467654937121908 p^{9} T^{12} - 1297704122709412 p^{12} T^{13} + 2107526963381 p^{15} T^{14} - 2746264024 p^{18} T^{15} + 2899914 p^{21} T^{16} - 2092 p^{24} T^{17} + p^{27} T^{18} \) | |
| 67 | \( 1 - 252 T + 1344218 T^{2} - 194838062 T^{3} + 924270634769 T^{4} - 62929997943194 T^{5} + 444172040315497378 T^{6} - 12691634394172178592 T^{7} + \)\(16\!\cdots\!42\)\( T^{8} - \)\(27\!\cdots\!68\)\( T^{9} + \)\(16\!\cdots\!42\)\( p^{3} T^{10} - 12691634394172178592 p^{6} T^{11} + 444172040315497378 p^{9} T^{12} - 62929997943194 p^{12} T^{13} + 924270634769 p^{15} T^{14} - 194838062 p^{18} T^{15} + 1344218 p^{21} T^{16} - 252 p^{24} T^{17} + p^{27} T^{18} \) | |
| 71 | \( 1 - 2352 T + 3882653 T^{2} - 4853291470 T^{3} + 5291140605433 T^{4} - 4946739246219630 T^{5} + 4165028619883159322 T^{6} - \)\(31\!\cdots\!84\)\( T^{7} + \)\(21\!\cdots\!45\)\( T^{8} - \)\(13\!\cdots\!48\)\( T^{9} + \)\(21\!\cdots\!45\)\( p^{3} T^{10} - \)\(31\!\cdots\!84\)\( p^{6} T^{11} + 4165028619883159322 p^{9} T^{12} - 4946739246219630 p^{12} T^{13} + 5291140605433 p^{15} T^{14} - 4853291470 p^{18} T^{15} + 3882653 p^{21} T^{16} - 2352 p^{24} T^{17} + p^{27} T^{18} \) | |
| 73 | \( 1 - 903 T + 2121557 T^{2} - 1395135673 T^{3} + 2097360891107 T^{4} - 1120858841873713 T^{5} + 1372229520080077430 T^{6} - \)\(62\!\cdots\!27\)\( T^{7} + \)\(67\!\cdots\!77\)\( T^{8} - \)\(27\!\cdots\!53\)\( T^{9} + \)\(67\!\cdots\!77\)\( p^{3} T^{10} - \)\(62\!\cdots\!27\)\( p^{6} T^{11} + 1372229520080077430 p^{9} T^{12} - 1120858841873713 p^{12} T^{13} + 2097360891107 p^{15} T^{14} - 1395135673 p^{18} T^{15} + 2121557 p^{21} T^{16} - 903 p^{24} T^{17} + p^{27} T^{18} \) | |
| 79 | \( 1 + 115 T + 1593876 T^{2} + 798800090 T^{3} + 1772916753533 T^{4} + 999425306482225 T^{5} + 1431928934682659048 T^{6} + \)\(86\!\cdots\!41\)\( T^{7} + \)\(91\!\cdots\!76\)\( T^{8} + \)\(47\!\cdots\!49\)\( T^{9} + \)\(91\!\cdots\!76\)\( p^{3} T^{10} + \)\(86\!\cdots\!41\)\( p^{6} T^{11} + 1431928934682659048 p^{9} T^{12} + 999425306482225 p^{12} T^{13} + 1772916753533 p^{15} T^{14} + 798800090 p^{18} T^{15} + 1593876 p^{21} T^{16} + 115 p^{24} T^{17} + p^{27} T^{18} \) | |
| 83 | \( 1 - 1207 T + 4131167 T^{2} - 3987000055 T^{3} + 7475399132903 T^{4} - 6018839061121269 T^{5} + 8136124359911725698 T^{6} - \)\(56\!\cdots\!47\)\( T^{7} + \)\(61\!\cdots\!63\)\( T^{8} - \)\(37\!\cdots\!81\)\( T^{9} + \)\(61\!\cdots\!63\)\( p^{3} T^{10} - \)\(56\!\cdots\!47\)\( p^{6} T^{11} + 8136124359911725698 p^{9} T^{12} - 6018839061121269 p^{12} T^{13} + 7475399132903 p^{15} T^{14} - 3987000055 p^{18} T^{15} + 4131167 p^{21} T^{16} - 1207 p^{24} T^{17} + p^{27} T^{18} \) | |
| 89 | \( 1 - 2336 T + 6327764 T^{2} - 10560197840 T^{3} + 17240363463629 T^{4} - 22342940548593066 T^{5} + 27449719261712183466 T^{6} - \)\(28\!\cdots\!50\)\( T^{7} + \)\(28\!\cdots\!84\)\( T^{8} - \)\(24\!\cdots\!88\)\( T^{9} + \)\(28\!\cdots\!84\)\( p^{3} T^{10} - \)\(28\!\cdots\!50\)\( p^{6} T^{11} + 27449719261712183466 p^{9} T^{12} - 22342940548593066 p^{12} T^{13} + 17240363463629 p^{15} T^{14} - 10560197840 p^{18} T^{15} + 6327764 p^{21} T^{16} - 2336 p^{24} T^{17} + p^{27} T^{18} \) | |
| 97 | \( 1 - 2155 T + 8642140 T^{2} - 14089662470 T^{3} + 31790566482928 T^{4} - 41403901430606002 T^{5} + 67281361883749063883 T^{6} - \)\(71\!\cdots\!76\)\( T^{7} + \)\(91\!\cdots\!12\)\( T^{8} - \)\(80\!\cdots\!81\)\( T^{9} + \)\(91\!\cdots\!12\)\( p^{3} T^{10} - \)\(71\!\cdots\!76\)\( p^{6} T^{11} + 67281361883749063883 p^{9} T^{12} - 41403901430606002 p^{12} T^{13} + 31790566482928 p^{15} T^{14} - 14089662470 p^{18} T^{15} + 8642140 p^{21} T^{16} - 2155 p^{24} T^{17} + p^{27} T^{18} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.05047499629271399839897108139, −3.97619123806160646577536968803, −3.90288222481510906834762943886, −3.87627373910893986795959781263, −3.75764532448328809261617092098, −3.61726439442384530100911359728, −3.54921232293870494391195718479, −3.04957979983645306198235924513, −2.75174472206153401029179723825, −2.44011343263255999129428323652, −2.40512302146340762385658018354, −2.34522878695574619282457823886, −2.25397976965333329948256947048, −2.09547075524925411666245668846, −1.85350484237476979473755534740, −1.80303940809505988019019324902, −1.74109481815853732590797028763, −1.38731984103624258405838410421, −1.24868050882996104966236815137, −0.819570935988362521911698726099, −0.67410225259641778215507783515, −0.65879724760744432710805663340, −0.59098278329328222635370691920, −0.44352378685980362385502198572, −0.41634657336465688632166560629, 0.41634657336465688632166560629, 0.44352378685980362385502198572, 0.59098278329328222635370691920, 0.65879724760744432710805663340, 0.67410225259641778215507783515, 0.819570935988362521911698726099, 1.24868050882996104966236815137, 1.38731984103624258405838410421, 1.74109481815853732590797028763, 1.80303940809505988019019324902, 1.85350484237476979473755534740, 2.09547075524925411666245668846, 2.25397976965333329948256947048, 2.34522878695574619282457823886, 2.40512302146340762385658018354, 2.44011343263255999129428323652, 2.75174472206153401029179723825, 3.04957979983645306198235924513, 3.54921232293870494391195718479, 3.61726439442384530100911359728, 3.75764532448328809261617092098, 3.87627373910893986795959781263, 3.90288222481510906834762943886, 3.97619123806160646577536968803, 4.05047499629271399839897108139
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.