Dirichlet series
| L(s) = 1 | − 8·2-s + 12·4-s − 41·5-s + 7-s + 59·8-s + 328·10-s − 37·11-s − 8·14-s − 200·16-s + 134·17-s − 72·19-s − 492·20-s + 296·22-s − 226·23-s + 584·25-s + 12·28-s + 547·29-s − 521·31-s + 127·32-s − 1.07e3·34-s − 41·35-s + 584·37-s + 576·38-s − 2.41e3·40-s − 482·41-s + 158·43-s − 444·44-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 3/2·4-s − 3.66·5-s + 0.0539·7-s + 2.60·8-s + 10.3·10-s − 1.01·11-s − 0.152·14-s − 3.12·16-s + 1.91·17-s − 0.869·19-s − 5.50·20-s + 2.86·22-s − 2.04·23-s + 4.67·25-s + 0.0809·28-s + 3.50·29-s − 3.01·31-s + 0.701·32-s − 5.40·34-s − 0.198·35-s + 2.59·37-s + 2.45·38-s − 9.56·40-s − 1.83·41-s + 0.560·43-s − 1.52·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 13^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 13^{18}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(3^{18} \cdot 13^{18}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(3.77535\times 10^{17}\) |
| Root analytic conductor: | \(9.47322\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 3^{18} \cdot 13^{18} ,\ ( \ : [3/2]^{9} ),\ -1 )\) |
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 | 3 | \( 1 \) |
| 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
−3.68501354441774100824149488695, −3.61787832365744119953567555673, −3.55738225100763759149951783788, −3.47543440708650300666939807958, −3.24720738460991646960055060941, −3.22974633133281156483619283858, −3.12327290638370777138077104626, −3.12200961742351463194891151350, −2.91092149370554596828966998316, −2.75642390804444750590536711255, −2.72108652912405512798394683595, −2.38015933451947247395429919962, −2.25624462383218863338117684689, −2.20634355613184297328335247062, −2.04809871847129013814105246116, −1.96456926305919571538432275391, −1.70480113990484922884241749053, −1.64014366068712475396512350575, −1.33445962347392965942214406102, −1.31268941788081637290527637344, −1.15918264399876418770142977721, −1.08135716005055182545664056525, −0.932744334486424127369807135665, −0.899851130727178962054585234175, −0.62903606801859920582642816113, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.62903606801859920582642816113, 0.899851130727178962054585234175, 0.932744334486424127369807135665, 1.08135716005055182545664056525, 1.15918264399876418770142977721, 1.31268941788081637290527637344, 1.33445962347392965942214406102, 1.64014366068712475396512350575, 1.70480113990484922884241749053, 1.96456926305919571538432275391, 2.04809871847129013814105246116, 2.20634355613184297328335247062, 2.25624462383218863338117684689, 2.38015933451947247395429919962, 2.72108652912405512798394683595, 2.75642390804444750590536711255, 2.91092149370554596828966998316, 3.12200961742351463194891151350, 3.12327290638370777138077104626, 3.22974633133281156483619283858, 3.24720738460991646960055060941, 3.47543440708650300666939807958, 3.55738225100763759149951783788, 3.61787832365744119953567555673, 3.68501354441774100824149488695
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.