Dirichlet series
| L(s) = 1 | − 5·2-s − 5·4-s − 30·5-s + 38·7-s + 60·8-s + 150·10-s − 181·11-s − 190·14-s − 13·16-s + 55·17-s + 161·19-s + 150·20-s + 905·22-s + 204·23-s + 41·25-s − 190·28-s − 280·29-s + 706·31-s − 400·32-s − 275·34-s − 1.14e3·35-s + 298·37-s − 805·38-s − 1.80e3·40-s − 1.20e3·41-s − 533·43-s + 905·44-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 5/8·4-s − 2.68·5-s + 2.05·7-s + 2.65·8-s + 4.74·10-s − 4.96·11-s − 3.62·14-s − 0.203·16-s + 0.784·17-s + 1.94·19-s + 1.67·20-s + 8.77·22-s + 1.84·23-s + 0.327·25-s − 1.28·28-s − 1.79·29-s + 4.09·31-s − 2.20·32-s − 1.38·34-s − 5.50·35-s + 1.32·37-s − 3.43·38-s − 7.11·40-s − 4.57·41-s − 1.89·43-s + 3.10·44-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 | \( 1 + 5 T + 15 p T^{2} + 115 T^{3} + 219 p T^{4} + 715 p T^{5} + 4743 T^{6} + 7291 p T^{7} + 1431 p^{5} T^{8} + 15927 p^{3} T^{9} + 1431 p^{8} T^{10} + 7291 p^{7} T^{11} + 4743 p^{9} T^{12} + 715 p^{13} T^{13} + 219 p^{16} T^{14} + 115 p^{18} T^{15} + 15 p^{22} T^{16} + 5 p^{24} T^{17} + p^{27} T^{18} \) |
| 5 | \( 1 + 6 p T + 859 T^{2} + 17314 T^{3} + 340943 T^{4} + 5555612 T^{5} + 85166304 T^{6} + 1151915472 T^{7} + 14768999549 T^{8} + 170409744152 T^{9} + 14768999549 p^{3} T^{10} + 1151915472 p^{6} T^{11} + 85166304 p^{9} T^{12} + 5555612 p^{12} T^{13} + 340943 p^{15} T^{14} + 17314 p^{18} T^{15} + 859 p^{21} T^{16} + 6 p^{25} T^{17} + p^{27} T^{18} \) | |
| 7 | \( 1 - 38 T + 2064 T^{2} - 55116 T^{3} + 1826925 T^{4} - 39304388 T^{5} + 1024949964 T^{6} - 18878435944 T^{7} + 426393254790 T^{8} - 7070968505084 T^{9} + 426393254790 p^{3} T^{10} - 18878435944 p^{6} T^{11} + 1024949964 p^{9} T^{12} - 39304388 p^{12} T^{13} + 1826925 p^{15} T^{14} - 55116 p^{18} T^{15} + 2064 p^{21} T^{16} - 38 p^{24} T^{17} + p^{27} T^{18} \) | |
| 11 | \( 1 + 181 T + 22813 T^{2} + 2092553 T^{3} + 158872467 T^{4} + 10131074403 T^{5} + 562968231928 T^{6} + 27429571614039 T^{7} + 1188788675103567 T^{8} + 45854313192147277 T^{9} + 1188788675103567 p^{3} T^{10} + 27429571614039 p^{6} T^{11} + 562968231928 p^{9} T^{12} + 10131074403 p^{12} T^{13} + 158872467 p^{15} T^{14} + 2092553 p^{18} T^{15} + 22813 p^{21} T^{16} + 181 p^{24} T^{17} + p^{27} T^{18} \) | |
| 17 | \( 1 - 55 T + 20781 T^{2} - 1015605 T^{3} + 210860987 T^{4} - 10154004217 T^{5} + 1490729786258 T^{6} - 74874380602311 T^{7} + 8516895972920457 T^{8} - 422323716428354157 T^{9} + 8516895972920457 p^{3} T^{10} - 74874380602311 p^{6} T^{11} + 1490729786258 p^{9} T^{12} - 10154004217 p^{12} T^{13} + 210860987 p^{15} T^{14} - 1015605 p^{18} T^{15} + 20781 p^{21} T^{16} - 55 p^{24} T^{17} + p^{27} T^{18} \) | |
| 19 | \( 1 - 161 T + 50439 T^{2} - 353563 p T^{3} + 1172104443 T^{4} - 131991570375 T^{5} + 16648090685398 T^{6} - 1600734207609033 T^{7} + 160884768934181395 T^{8} - 13163166447747413447 T^{9} + 160884768934181395 p^{3} T^{10} - 1600734207609033 p^{6} T^{11} + 16648090685398 p^{9} T^{12} - 131991570375 p^{12} T^{13} + 1172104443 p^{15} T^{14} - 353563 p^{19} T^{15} + 50439 p^{21} T^{16} - 161 p^{24} T^{17} + p^{27} T^{18} \) | |
| 23 | \( 1 - 204 T + 80372 T^{2} - 14448880 T^{3} + 3205235016 T^{4} - 492290427780 T^{5} + 80711192199761 T^{6} - 10490406488019272 T^{7} + 1391371628570010740 T^{8} - \)\(15\!\cdots\!12\)\( T^{9} + 1391371628570010740 p^{3} T^{10} - 10490406488019272 p^{6} T^{11} + 80711192199761 p^{9} T^{12} - 492290427780 p^{12} T^{13} + 3205235016 p^{15} T^{14} - 14448880 p^{18} T^{15} + 80372 p^{21} T^{16} - 204 p^{24} T^{17} + p^{27} T^{18} \) | |
| 29 | \( 1 + 280 T + 170966 T^{2} + 42063240 T^{3} + 14103634381 T^{4} + 2956561889248 T^{5} + 724380432808236 T^{6} + 128184309066850350 T^{7} + 25237920158245437728 T^{8} + \)\(37\!\cdots\!48\)\( T^{9} + 25237920158245437728 p^{3} T^{10} + 128184309066850350 p^{6} T^{11} + 724380432808236 p^{9} T^{12} + 2956561889248 p^{12} T^{13} + 14103634381 p^{15} T^{14} + 42063240 p^{18} T^{15} + 170966 p^{21} T^{16} + 280 p^{24} T^{17} + p^{27} T^{18} \) | |
| 31 | \( 1 - 706 T + 416348 T^{2} - 167129464 T^{3} + 1911356387 p T^{4} - 17113845091756 T^{5} + 4471497849821796 T^{6} - 1005049239010198696 T^{7} + \)\(20\!\cdots\!78\)\( T^{8} - \)\(37\!\cdots\!00\)\( T^{9} + \)\(20\!\cdots\!78\)\( p^{3} T^{10} - 1005049239010198696 p^{6} T^{11} + 4471497849821796 p^{9} T^{12} - 17113845091756 p^{12} T^{13} + 1911356387 p^{16} T^{14} - 167129464 p^{18} T^{15} + 416348 p^{21} T^{16} - 706 p^{24} T^{17} + p^{27} T^{18} \) | |
| 37 | \( 1 - 298 T + 223779 T^{2} - 32817014 T^{3} + 18047849351 T^{4} - 975487786900 T^{5} + 1025783321374520 T^{6} - 13632457613223580 T^{7} + 1615763381169657697 p T^{8} - \)\(67\!\cdots\!12\)\( T^{9} + 1615763381169657697 p^{4} T^{10} - 13632457613223580 p^{6} T^{11} + 1025783321374520 p^{9} T^{12} - 975487786900 p^{12} T^{13} + 18047849351 p^{15} T^{14} - 32817014 p^{18} T^{15} + 223779 p^{21} T^{16} - 298 p^{24} T^{17} + p^{27} T^{18} \) | |
| 41 | \( 1 + 1201 T + 1114424 T^{2} + 719014126 T^{3} + 395257426664 T^{4} + 178581306674998 T^{5} + 71496675012390615 T^{6} + 24782835331514842776 T^{7} + \)\(77\!\cdots\!32\)\( T^{8} + \)\(21\!\cdots\!19\)\( T^{9} + \)\(77\!\cdots\!32\)\( p^{3} T^{10} + 24782835331514842776 p^{6} T^{11} + 71496675012390615 p^{9} T^{12} + 178581306674998 p^{12} T^{13} + 395257426664 p^{15} T^{14} + 719014126 p^{18} T^{15} + 1114424 p^{21} T^{16} + 1201 p^{24} T^{17} + p^{27} T^{18} \) | |
| 43 | \( 1 + 533 T + 562484 T^{2} + 202194824 T^{3} + 127846667481 T^{4} + 34776686916413 T^{5} + 17528418199113624 T^{6} + 3913634689613466129 T^{7} + \)\(17\!\cdots\!12\)\( T^{8} + \)\(34\!\cdots\!53\)\( T^{9} + \)\(17\!\cdots\!12\)\( p^{3} T^{10} + 3913634689613466129 p^{6} T^{11} + 17528418199113624 p^{9} T^{12} + 34776686916413 p^{12} T^{13} + 127846667481 p^{15} T^{14} + 202194824 p^{18} T^{15} + 562484 p^{21} T^{16} + 533 p^{24} T^{17} + p^{27} T^{18} \) | |
| 47 | \( 1 + 956 T + 1071019 T^{2} + 666300230 T^{3} + 434228172393 T^{4} + 202420083652096 T^{5} + 97312087273615934 T^{6} + 36574221973341845292 T^{7} + \)\(14\!\cdots\!23\)\( T^{8} + \)\(44\!\cdots\!76\)\( T^{9} + \)\(14\!\cdots\!23\)\( p^{3} T^{10} + 36574221973341845292 p^{6} T^{11} + 97312087273615934 p^{9} T^{12} + 202420083652096 p^{12} T^{13} + 434228172393 p^{15} T^{14} + 666300230 p^{18} T^{15} + 1071019 p^{21} T^{16} + 956 p^{24} T^{17} + p^{27} T^{18} \) | |
| 53 | \( 1 - 278 T + 773518 T^{2} - 207620330 T^{3} + 319892405009 T^{4} - 77727314367624 T^{5} + 87908254531904768 T^{6} - 19135768589714679110 T^{7} + \)\(17\!\cdots\!16\)\( T^{8} - \)\(33\!\cdots\!56\)\( T^{9} + \)\(17\!\cdots\!16\)\( p^{3} T^{10} - 19135768589714679110 p^{6} T^{11} + 87908254531904768 p^{9} T^{12} - 77727314367624 p^{12} T^{13} + 319892405009 p^{15} T^{14} - 207620330 p^{18} T^{15} + 773518 p^{21} T^{16} - 278 p^{24} T^{17} + p^{27} T^{18} \) | |
| 59 | \( 1 + 1377 T + 35616 p T^{2} + 1890354682 T^{3} + 1677650029688 T^{4} + 1137678842611544 T^{5} + 744346433078864313 T^{6} + \)\(40\!\cdots\!76\)\( T^{7} + \)\(21\!\cdots\!48\)\( T^{8} + \)\(99\!\cdots\!45\)\( T^{9} + \)\(21\!\cdots\!48\)\( p^{3} T^{10} + \)\(40\!\cdots\!76\)\( p^{6} T^{11} + 744346433078864313 p^{9} T^{12} + 1137678842611544 p^{12} T^{13} + 1677650029688 p^{15} T^{14} + 1890354682 p^{18} T^{15} + 35616 p^{22} T^{16} + 1377 p^{24} T^{17} + p^{27} T^{18} \) | |
| 61 | \( 1 + 136 T + 911823 T^{2} + 59406622 T^{3} + 460946379119 T^{4} + 30738075515466 T^{5} + 173448031060568832 T^{6} + 12174546108023808910 T^{7} + \)\(48\!\cdots\!85\)\( T^{8} + \)\(29\!\cdots\!28\)\( T^{9} + \)\(48\!\cdots\!85\)\( p^{3} T^{10} + 12174546108023808910 p^{6} T^{11} + 173448031060568832 p^{9} T^{12} + 30738075515466 p^{12} T^{13} + 460946379119 p^{15} T^{14} + 59406622 p^{18} T^{15} + 911823 p^{21} T^{16} + 136 p^{24} T^{17} + p^{27} T^{18} \) | |
| 67 | \( 1 + 931 T + 1565532 T^{2} + 1349146552 T^{3} + 1348667779325 T^{4} + 984130862654187 T^{5} + 769662941782037760 T^{6} + \)\(48\!\cdots\!75\)\( T^{7} + \)\(31\!\cdots\!20\)\( T^{8} + \)\(16\!\cdots\!51\)\( T^{9} + \)\(31\!\cdots\!20\)\( p^{3} T^{10} + \)\(48\!\cdots\!75\)\( p^{6} T^{11} + 769662941782037760 p^{9} T^{12} + 984130862654187 p^{12} T^{13} + 1348667779325 p^{15} T^{14} + 1349146552 p^{18} T^{15} + 1565532 p^{21} T^{16} + 931 p^{24} T^{17} + p^{27} T^{18} \) | |
| 71 | \( 1 + 2046 T + 3529486 T^{2} + 4409681886 T^{3} + 4949231088207 T^{4} + 4633134906681946 T^{5} + 3980764105294282426 T^{6} + \)\(30\!\cdots\!06\)\( T^{7} + \)\(20\!\cdots\!28\)\( T^{8} + \)\(12\!\cdots\!32\)\( T^{9} + \)\(20\!\cdots\!28\)\( p^{3} T^{10} + \)\(30\!\cdots\!06\)\( p^{6} T^{11} + 3980764105294282426 p^{9} T^{12} + 4633134906681946 p^{12} T^{13} + 4949231088207 p^{15} T^{14} + 4409681886 p^{18} T^{15} + 3529486 p^{21} T^{16} + 2046 p^{24} T^{17} + p^{27} T^{18} \) | |
| 73 | \( 1 + 45 T + 1891466 T^{2} - 33830186 T^{3} + 1674355114259 T^{4} - 33519963542817 T^{5} + 983076362313015860 T^{6} + 18994886524736826951 T^{7} + \)\(45\!\cdots\!72\)\( T^{8} + \)\(18\!\cdots\!55\)\( T^{9} + \)\(45\!\cdots\!72\)\( p^{3} T^{10} + 18994886524736826951 p^{6} T^{11} + 983076362313015860 p^{9} T^{12} - 33519963542817 p^{12} T^{13} + 1674355114259 p^{15} T^{14} - 33830186 p^{18} T^{15} + 1891466 p^{21} T^{16} + 45 p^{24} T^{17} + p^{27} T^{18} \) | |
| 79 | \( 1 - 412 T + 3404882 T^{2} - 1342603600 T^{3} + 5422429448055 T^{4} - 2004760282589858 T^{5} + 5353334618487262810 T^{6} - \)\(18\!\cdots\!82\)\( T^{7} + \)\(36\!\cdots\!76\)\( T^{8} - \)\(10\!\cdots\!88\)\( T^{9} + \)\(36\!\cdots\!76\)\( p^{3} T^{10} - \)\(18\!\cdots\!82\)\( p^{6} T^{11} + 5353334618487262810 p^{9} T^{12} - 2004760282589858 p^{12} T^{13} + 5422429448055 p^{15} T^{14} - 1342603600 p^{18} T^{15} + 3404882 p^{21} T^{16} - 412 p^{24} T^{17} + p^{27} T^{18} \) | |
| 83 | \( 1 + 3709 T + 9253460 T^{2} + 15842855210 T^{3} + 21895132541837 T^{4} + 24308978625331839 T^{5} + 23407062219850258008 T^{6} + \)\(19\!\cdots\!91\)\( T^{7} + \)\(15\!\cdots\!88\)\( T^{8} + \)\(11\!\cdots\!75\)\( T^{9} + \)\(15\!\cdots\!88\)\( p^{3} T^{10} + \)\(19\!\cdots\!91\)\( p^{6} T^{11} + 23407062219850258008 p^{9} T^{12} + 24308978625331839 p^{12} T^{13} + 21895132541837 p^{15} T^{14} + 15842855210 p^{18} T^{15} + 9253460 p^{21} T^{16} + 3709 p^{24} T^{17} + p^{27} T^{18} \) | |
| 89 | \( 1 + 1663 T + 6073458 T^{2} + 7244833042 T^{3} + 15263017405431 T^{4} + 14374017171712377 T^{5} + 22652055451533915976 T^{6} + \)\(17\!\cdots\!45\)\( T^{7} + \)\(22\!\cdots\!36\)\( T^{8} + \)\(14\!\cdots\!49\)\( T^{9} + \)\(22\!\cdots\!36\)\( p^{3} T^{10} + \)\(17\!\cdots\!45\)\( p^{6} T^{11} + 22652055451533915976 p^{9} T^{12} + 14374017171712377 p^{12} T^{13} + 15263017405431 p^{15} T^{14} + 7244833042 p^{18} T^{15} + 6073458 p^{21} T^{16} + 1663 p^{24} T^{17} + p^{27} T^{18} \) | |
| 97 | \( 1 + 1087 T + 5134886 T^{2} + 5981480330 T^{3} + 13805202760179 T^{4} + 15130273407572117 T^{5} + 24627649618671936284 T^{6} + \)\(23\!\cdots\!85\)\( T^{7} + \)\(31\!\cdots\!32\)\( T^{8} + \)\(25\!\cdots\!21\)\( T^{9} + \)\(31\!\cdots\!32\)\( p^{3} T^{10} + \)\(23\!\cdots\!85\)\( p^{6} T^{11} + 24627649618671936284 p^{9} T^{12} + 15130273407572117 p^{12} T^{13} + 13805202760179 p^{15} T^{14} + 5981480330 p^{18} T^{15} + 5134886 p^{21} T^{16} + 1087 p^{24} T^{17} + p^{27} T^{18} \) | |
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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
−3.60179437977633171083075077373, −3.53872048296578636291659110660, −3.53434235964978106919470152521, −3.38969437614766216918402784780, −3.32851572485504237770231940900, −3.14751156106323545303250138526, −3.10005348422326055369122960012, −3.03456208747403410637679088548, −2.94712342997024654682628506190, −2.75392345997472372101979916403, −2.71916906734690053454793411819, −2.44865503704240050320960014807, −2.43065611719649612950569119513, −2.21995807984764288790155022213, −2.07866221704800265486597120908, −1.89653200096778406652831147654, −1.69708801039797298027376255245, −1.64690387123538852668268591512, −1.43790667312561703860417595574, −1.33591363340096266766889457586, −1.29747420248327725709859458532, −1.10368376252315050743894441348, −1.06028424866145636400372665535, −0.890047844908469598145731745895, −0.75970020384644945356497046233, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.75970020384644945356497046233, 0.890047844908469598145731745895, 1.06028424866145636400372665535, 1.10368376252315050743894441348, 1.29747420248327725709859458532, 1.33591363340096266766889457586, 1.43790667312561703860417595574, 1.64690387123538852668268591512, 1.69708801039797298027376255245, 1.89653200096778406652831147654, 2.07866221704800265486597120908, 2.21995807984764288790155022213, 2.43065611719649612950569119513, 2.44865503704240050320960014807, 2.71916906734690053454793411819, 2.75392345997472372101979916403, 2.94712342997024654682628506190, 3.03456208747403410637679088548, 3.10005348422326055369122960012, 3.14751156106323545303250138526, 3.32851572485504237770231940900, 3.38969437614766216918402784780, 3.53434235964978106919470152521, 3.53872048296578636291659110660, 3.60179437977633171083075077373
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.