Dirichlet series
| L(s) = 1 | − 32·4-s − 20·5-s − 119·7-s − 54·11-s − 66·13-s + 475·16-s − 68·17-s − 26·19-s + 640·20-s + 391·23-s − 568·25-s + 3.80e3·28-s − 24·29-s + 4·31-s − 18·32-s + 2.38e3·35-s − 490·37-s − 480·41-s − 504·43-s + 1.72e3·44-s − 1.34e3·47-s + 7.49e3·49-s + 2.11e3·52-s − 366·53-s + 1.08e3·55-s − 2.95e3·59-s − 1.10e3·61-s + ⋯ |
| L(s) = 1 | − 4·4-s − 1.78·5-s − 6.42·7-s − 1.48·11-s − 1.40·13-s + 7.42·16-s − 0.970·17-s − 0.313·19-s + 7.15·20-s + 3.54·23-s − 4.54·25-s + 25.7·28-s − 0.153·29-s + 0.0231·31-s − 0.0994·32-s + 11.4·35-s − 2.17·37-s − 1.82·41-s − 1.78·43-s + 5.92·44-s − 4.15·47-s + 21.8·49-s + 5.63·52-s − 0.948·53-s + 2.64·55-s − 6.52·59-s − 2.31·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{34} \cdot 7^{17} \cdot 23^{17}\right)^{s/2} \, \Gamma_{\C}(s)^{17} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{34} \cdot 7^{17} \cdot 23^{17}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{17} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(34\) |
| Conductor: | \(3^{34} \cdot 7^{17} \cdot 23^{17}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(6.96443\times 10^{32}\) |
| Root analytic conductor: | \(9.24628\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(17\) |
| Selberg data: | \((34,\ 3^{34} \cdot 7^{17} \cdot 23^{17} ,\ ( \ : [3/2]^{17} ),\ -1 )\) |
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 | \( 1 \) |
| 7 | \( ( 1 + p T )^{17} \) | |
| 23 | \( ( 1 - p T )^{17} \) | |
| good | 2 | \( 1 + p^{5} T^{2} + 549 T^{4} + 9 p T^{5} + 6741 T^{6} + 479 p T^{7} + 16455 p^{2} T^{8} + 2187 p T^{9} + 531511 T^{10} - 100059 p T^{11} + 1827489 p T^{12} - 298913 p^{4} T^{13} + 1434927 p^{4} T^{14} - 1947435 p^{5} T^{15} + 4599781 p^{5} T^{16} - 563119 p^{10} T^{17} + 4599781 p^{8} T^{18} - 1947435 p^{11} T^{19} + 1434927 p^{13} T^{20} - 298913 p^{16} T^{21} + 1827489 p^{16} T^{22} - 100059 p^{19} T^{23} + 531511 p^{21} T^{24} + 2187 p^{25} T^{25} + 16455 p^{29} T^{26} + 479 p^{31} T^{27} + 6741 p^{33} T^{28} + 9 p^{37} T^{29} + 549 p^{39} T^{30} + p^{50} T^{32} + p^{51} T^{34} \) |
| 5 | \( 1 + 4 p T + 968 T^{2} + 17412 T^{3} + 479916 T^{4} + 7640332 T^{5} + 31191919 p T^{6} + 2200983828 T^{7} + 36568174931 T^{8} + 91732971152 p T^{9} + 6482999190974 T^{10} + 72611787306244 T^{11} + 179748102167152 p T^{12} + 9113678918521788 T^{13} + 102926745204639937 T^{14} + 991973016961739268 T^{15} + 11157144119487515862 T^{16} + 22623055215379079368 p T^{17} + 11157144119487515862 p^{3} T^{18} + 991973016961739268 p^{6} T^{19} + 102926745204639937 p^{9} T^{20} + 9113678918521788 p^{12} T^{21} + 179748102167152 p^{16} T^{22} + 72611787306244 p^{18} T^{23} + 6482999190974 p^{21} T^{24} + 91732971152 p^{25} T^{25} + 36568174931 p^{27} T^{26} + 2200983828 p^{30} T^{27} + 31191919 p^{34} T^{28} + 7640332 p^{36} T^{29} + 479916 p^{39} T^{30} + 17412 p^{42} T^{31} + 968 p^{45} T^{32} + 4 p^{49} T^{33} + p^{51} T^{34} \) | |
| 11 | \( 1 + 54 T + 11502 T^{2} + 539480 T^{3} + 63776727 T^{4} + 2521712980 T^{5} + 221610563701 T^{6} + 7085120529392 T^{7} + 527854386874874 T^{8} + 12315475348785326 T^{9} + 879663416964577059 T^{10} + 9887685506681654208 T^{11} + \)\(97\!\cdots\!50\)\( T^{12} - \)\(11\!\cdots\!88\)\( T^{13} + \)\(54\!\cdots\!08\)\( T^{14} - \)\(54\!\cdots\!60\)\( T^{15} - \)\(15\!\cdots\!58\)\( T^{16} - \)\(95\!\cdots\!24\)\( T^{17} - \)\(15\!\cdots\!58\)\( p^{3} T^{18} - \)\(54\!\cdots\!60\)\( p^{6} T^{19} + \)\(54\!\cdots\!08\)\( p^{9} T^{20} - \)\(11\!\cdots\!88\)\( p^{12} T^{21} + \)\(97\!\cdots\!50\)\( p^{15} T^{22} + 9887685506681654208 p^{18} T^{23} + 879663416964577059 p^{21} T^{24} + 12315475348785326 p^{24} T^{25} + 527854386874874 p^{27} T^{26} + 7085120529392 p^{30} T^{27} + 221610563701 p^{33} T^{28} + 2521712980 p^{36} T^{29} + 63776727 p^{39} T^{30} + 539480 p^{42} T^{31} + 11502 p^{45} T^{32} + 54 p^{48} T^{33} + p^{51} T^{34} \) | |
| 13 | \( 1 + 66 T + 22660 T^{2} + 1364914 T^{3} + 255798216 T^{4} + 14241976022 T^{5} + 1921476926679 T^{6} + 99470375757904 T^{7} + 10773912383242363 T^{8} + 519983996953528526 T^{9} + 47855053022401657842 T^{10} + \)\(21\!\cdots\!98\)\( T^{11} + \)\(17\!\cdots\!16\)\( T^{12} + \)\(73\!\cdots\!38\)\( T^{13} + \)\(53\!\cdots\!41\)\( T^{14} + \)\(20\!\cdots\!88\)\( T^{15} + \)\(13\!\cdots\!86\)\( T^{16} + \)\(49\!\cdots\!00\)\( T^{17} + \)\(13\!\cdots\!86\)\( p^{3} T^{18} + \)\(20\!\cdots\!88\)\( p^{6} T^{19} + \)\(53\!\cdots\!41\)\( p^{9} T^{20} + \)\(73\!\cdots\!38\)\( p^{12} T^{21} + \)\(17\!\cdots\!16\)\( p^{15} T^{22} + \)\(21\!\cdots\!98\)\( p^{18} T^{23} + 47855053022401657842 p^{21} T^{24} + 519983996953528526 p^{24} T^{25} + 10773912383242363 p^{27} T^{26} + 99470375757904 p^{30} T^{27} + 1921476926679 p^{33} T^{28} + 14241976022 p^{36} T^{29} + 255798216 p^{39} T^{30} + 1364914 p^{42} T^{31} + 22660 p^{45} T^{32} + 66 p^{48} T^{33} + p^{51} T^{34} \) | |
| 17 | \( 1 + 4 p T + 38883 T^{2} + 1813100 T^{3} + 684598124 T^{4} + 20987255244 T^{5} + 7514785166392 T^{6} + 145555102899984 T^{7} + 59844853193523919 T^{8} + 880197855720757848 T^{9} + \)\(38\!\cdots\!65\)\( T^{10} + \)\(74\!\cdots\!40\)\( T^{11} + \)\(21\!\cdots\!50\)\( T^{12} + \)\(70\!\cdots\!28\)\( T^{13} + \)\(10\!\cdots\!42\)\( T^{14} + \)\(53\!\cdots\!56\)\( T^{15} + \)\(51\!\cdots\!52\)\( T^{16} + \)\(30\!\cdots\!32\)\( T^{17} + \)\(51\!\cdots\!52\)\( p^{3} T^{18} + \)\(53\!\cdots\!56\)\( p^{6} T^{19} + \)\(10\!\cdots\!42\)\( p^{9} T^{20} + \)\(70\!\cdots\!28\)\( p^{12} T^{21} + \)\(21\!\cdots\!50\)\( p^{15} T^{22} + \)\(74\!\cdots\!40\)\( p^{18} T^{23} + \)\(38\!\cdots\!65\)\( p^{21} T^{24} + 880197855720757848 p^{24} T^{25} + 59844853193523919 p^{27} T^{26} + 145555102899984 p^{30} T^{27} + 7514785166392 p^{33} T^{28} + 20987255244 p^{36} T^{29} + 684598124 p^{39} T^{30} + 1813100 p^{42} T^{31} + 38883 p^{45} T^{32} + 4 p^{49} T^{33} + p^{51} T^{34} \) | |
| 19 | \( 1 + 26 T + 46064 T^{2} + 913868 T^{3} + 1092608139 T^{4} + 21546522884 T^{5} + 17707614428945 T^{6} + 409692687429116 T^{7} + 220576002673586492 T^{8} + 6268107491540020642 T^{9} + \)\(22\!\cdots\!15\)\( T^{10} + \)\(76\!\cdots\!64\)\( T^{11} + \)\(20\!\cdots\!02\)\( T^{12} + \)\(76\!\cdots\!00\)\( T^{13} + \)\(16\!\cdots\!72\)\( T^{14} + \)\(64\!\cdots\!44\)\( T^{15} + \)\(12\!\cdots\!50\)\( T^{16} + \)\(47\!\cdots\!28\)\( T^{17} + \)\(12\!\cdots\!50\)\( p^{3} T^{18} + \)\(64\!\cdots\!44\)\( p^{6} T^{19} + \)\(16\!\cdots\!72\)\( p^{9} T^{20} + \)\(76\!\cdots\!00\)\( p^{12} T^{21} + \)\(20\!\cdots\!02\)\( p^{15} T^{22} + \)\(76\!\cdots\!64\)\( p^{18} T^{23} + \)\(22\!\cdots\!15\)\( p^{21} T^{24} + 6268107491540020642 p^{24} T^{25} + 220576002673586492 p^{27} T^{26} + 409692687429116 p^{30} T^{27} + 17707614428945 p^{33} T^{28} + 21546522884 p^{36} T^{29} + 1092608139 p^{39} T^{30} + 913868 p^{42} T^{31} + 46064 p^{45} T^{32} + 26 p^{48} T^{33} + p^{51} T^{34} \) | |
| 29 | \( 1 + 24 T + 210368 T^{2} + 7306792 T^{3} + 22544186371 T^{4} + 1038008005592 T^{5} + 1638219079597563 T^{6} + 92142490502981712 T^{7} + 90687483163195277120 T^{8} + \)\(57\!\cdots\!52\)\( T^{9} + \)\(40\!\cdots\!65\)\( T^{10} + \)\(27\!\cdots\!52\)\( T^{11} + \)\(15\!\cdots\!74\)\( T^{12} + \)\(10\!\cdots\!20\)\( T^{13} + \)\(49\!\cdots\!36\)\( T^{14} + \)\(33\!\cdots\!52\)\( T^{15} + \)\(13\!\cdots\!62\)\( T^{16} + \)\(89\!\cdots\!44\)\( T^{17} + \)\(13\!\cdots\!62\)\( p^{3} T^{18} + \)\(33\!\cdots\!52\)\( p^{6} T^{19} + \)\(49\!\cdots\!36\)\( p^{9} T^{20} + \)\(10\!\cdots\!20\)\( p^{12} T^{21} + \)\(15\!\cdots\!74\)\( p^{15} T^{22} + \)\(27\!\cdots\!52\)\( p^{18} T^{23} + \)\(40\!\cdots\!65\)\( p^{21} T^{24} + \)\(57\!\cdots\!52\)\( p^{24} T^{25} + 90687483163195277120 p^{27} T^{26} + 92142490502981712 p^{30} T^{27} + 1638219079597563 p^{33} T^{28} + 1038008005592 p^{36} T^{29} + 22544186371 p^{39} T^{30} + 7306792 p^{42} T^{31} + 210368 p^{45} T^{32} + 24 p^{48} T^{33} + p^{51} T^{34} \) | |
| 31 | \( 1 - 4 T + 224117 T^{2} - 3894264 T^{3} + 25236823588 T^{4} - 777763278952 T^{5} + 1932602307089308 T^{6} - 85025259591950200 T^{7} + \)\(11\!\cdots\!91\)\( T^{8} - \)\(64\!\cdots\!40\)\( T^{9} + \)\(55\!\cdots\!35\)\( T^{10} - \)\(37\!\cdots\!76\)\( T^{11} + \)\(23\!\cdots\!58\)\( T^{12} - \)\(17\!\cdots\!04\)\( T^{13} + \)\(87\!\cdots\!82\)\( T^{14} - \)\(67\!\cdots\!24\)\( T^{15} + \)\(29\!\cdots\!72\)\( T^{16} - \)\(21\!\cdots\!08\)\( T^{17} + \)\(29\!\cdots\!72\)\( p^{3} T^{18} - \)\(67\!\cdots\!24\)\( p^{6} T^{19} + \)\(87\!\cdots\!82\)\( p^{9} T^{20} - \)\(17\!\cdots\!04\)\( p^{12} T^{21} + \)\(23\!\cdots\!58\)\( p^{15} T^{22} - \)\(37\!\cdots\!76\)\( p^{18} T^{23} + \)\(55\!\cdots\!35\)\( p^{21} T^{24} - \)\(64\!\cdots\!40\)\( p^{24} T^{25} + \)\(11\!\cdots\!91\)\( p^{27} T^{26} - 85025259591950200 p^{30} T^{27} + 1932602307089308 p^{33} T^{28} - 777763278952 p^{36} T^{29} + 25236823588 p^{39} T^{30} - 3894264 p^{42} T^{31} + 224117 p^{45} T^{32} - 4 p^{48} T^{33} + p^{51} T^{34} \) | |
| 37 | \( 1 + 490 T + 449394 T^{2} + 165105814 T^{3} + 89947401083 T^{4} + 25771196311492 T^{5} + 10704342653870831 T^{6} + 2428676672245807006 T^{7} + 23559909602505755146 p T^{8} + \)\(15\!\cdots\!38\)\( T^{9} + \)\(55\!\cdots\!49\)\( T^{10} + \)\(85\!\cdots\!60\)\( T^{11} + \)\(34\!\cdots\!90\)\( T^{12} + \)\(50\!\cdots\!56\)\( T^{13} + \)\(21\!\cdots\!08\)\( T^{14} + \)\(33\!\cdots\!48\)\( T^{15} + \)\(13\!\cdots\!78\)\( T^{16} + \)\(19\!\cdots\!00\)\( T^{17} + \)\(13\!\cdots\!78\)\( p^{3} T^{18} + \)\(33\!\cdots\!48\)\( p^{6} T^{19} + \)\(21\!\cdots\!08\)\( p^{9} T^{20} + \)\(50\!\cdots\!56\)\( p^{12} T^{21} + \)\(34\!\cdots\!90\)\( p^{15} T^{22} + \)\(85\!\cdots\!60\)\( p^{18} T^{23} + \)\(55\!\cdots\!49\)\( p^{21} T^{24} + \)\(15\!\cdots\!38\)\( p^{24} T^{25} + 23559909602505755146 p^{28} T^{26} + 2428676672245807006 p^{30} T^{27} + 10704342653870831 p^{33} T^{28} + 25771196311492 p^{36} T^{29} + 89947401083 p^{39} T^{30} + 165105814 p^{42} T^{31} + 449394 p^{45} T^{32} + 490 p^{48} T^{33} + p^{51} T^{34} \) | |
| 41 | \( 1 + 480 T + 589415 T^{2} + 231033496 T^{3} + 172739978369 T^{4} + 59583814895696 T^{5} + 34527053453887093 T^{6} + 10835958407378010472 T^{7} + \)\(52\!\cdots\!27\)\( T^{8} + \)\(15\!\cdots\!64\)\( T^{9} + \)\(66\!\cdots\!97\)\( T^{10} + \)\(17\!\cdots\!36\)\( T^{11} + \)\(69\!\cdots\!26\)\( T^{12} + \)\(17\!\cdots\!84\)\( T^{13} + \)\(62\!\cdots\!18\)\( T^{14} + \)\(14\!\cdots\!80\)\( T^{15} + \)\(49\!\cdots\!94\)\( T^{16} + \)\(11\!\cdots\!64\)\( T^{17} + \)\(49\!\cdots\!94\)\( p^{3} T^{18} + \)\(14\!\cdots\!80\)\( p^{6} T^{19} + \)\(62\!\cdots\!18\)\( p^{9} T^{20} + \)\(17\!\cdots\!84\)\( p^{12} T^{21} + \)\(69\!\cdots\!26\)\( p^{15} T^{22} + \)\(17\!\cdots\!36\)\( p^{18} T^{23} + \)\(66\!\cdots\!97\)\( p^{21} T^{24} + \)\(15\!\cdots\!64\)\( p^{24} T^{25} + \)\(52\!\cdots\!27\)\( p^{27} T^{26} + 10835958407378010472 p^{30} T^{27} + 34527053453887093 p^{33} T^{28} + 59583814895696 p^{36} T^{29} + 172739978369 p^{39} T^{30} + 231033496 p^{42} T^{31} + 589415 p^{45} T^{32} + 480 p^{48} T^{33} + p^{51} T^{34} \) | |
| 43 | \( 1 + 504 T + 905506 T^{2} + 381719048 T^{3} + 383904587966 T^{4} + 140384798944712 T^{5} + 102929926625623669 T^{6} + 33377959800723054424 T^{7} + \)\(19\!\cdots\!91\)\( T^{8} + \)\(57\!\cdots\!48\)\( T^{9} + \)\(29\!\cdots\!36\)\( T^{10} + \)\(77\!\cdots\!08\)\( T^{11} + \)\(34\!\cdots\!78\)\( T^{12} + \)\(86\!\cdots\!80\)\( T^{13} + \)\(35\!\cdots\!55\)\( T^{14} + \)\(81\!\cdots\!80\)\( T^{15} + \)\(31\!\cdots\!98\)\( T^{16} + \)\(68\!\cdots\!48\)\( T^{17} + \)\(31\!\cdots\!98\)\( p^{3} T^{18} + \)\(81\!\cdots\!80\)\( p^{6} T^{19} + \)\(35\!\cdots\!55\)\( p^{9} T^{20} + \)\(86\!\cdots\!80\)\( p^{12} T^{21} + \)\(34\!\cdots\!78\)\( p^{15} T^{22} + \)\(77\!\cdots\!08\)\( p^{18} T^{23} + \)\(29\!\cdots\!36\)\( p^{21} T^{24} + \)\(57\!\cdots\!48\)\( p^{24} T^{25} + \)\(19\!\cdots\!91\)\( p^{27} T^{26} + 33377959800723054424 p^{30} T^{27} + 102929926625623669 p^{33} T^{28} + 140384798944712 p^{36} T^{29} + 383904587966 p^{39} T^{30} + 381719048 p^{42} T^{31} + 905506 p^{45} T^{32} + 504 p^{48} T^{33} + p^{51} T^{34} \) | |
| 47 | \( 1 + 1340 T + 1621999 T^{2} + 1359940448 T^{3} + 1050181744417 T^{4} + 683747825992148 T^{5} + 417311281754436825 T^{6} + \)\(22\!\cdots\!28\)\( T^{7} + \)\(11\!\cdots\!44\)\( T^{8} + \)\(57\!\cdots\!92\)\( T^{9} + \)\(26\!\cdots\!04\)\( T^{10} + \)\(11\!\cdots\!12\)\( T^{11} + \)\(46\!\cdots\!61\)\( T^{12} + \)\(18\!\cdots\!40\)\( T^{13} + \)\(68\!\cdots\!61\)\( T^{14} + \)\(24\!\cdots\!72\)\( T^{15} + \)\(84\!\cdots\!10\)\( T^{16} + \)\(58\!\cdots\!92\)\( p T^{17} + \)\(84\!\cdots\!10\)\( p^{3} T^{18} + \)\(24\!\cdots\!72\)\( p^{6} T^{19} + \)\(68\!\cdots\!61\)\( p^{9} T^{20} + \)\(18\!\cdots\!40\)\( p^{12} T^{21} + \)\(46\!\cdots\!61\)\( p^{15} T^{22} + \)\(11\!\cdots\!12\)\( p^{18} T^{23} + \)\(26\!\cdots\!04\)\( p^{21} T^{24} + \)\(57\!\cdots\!92\)\( p^{24} T^{25} + \)\(11\!\cdots\!44\)\( p^{27} T^{26} + \)\(22\!\cdots\!28\)\( p^{30} T^{27} + 417311281754436825 p^{33} T^{28} + 683747825992148 p^{36} T^{29} + 1050181744417 p^{39} T^{30} + 1359940448 p^{42} T^{31} + 1621999 p^{45} T^{32} + 1340 p^{48} T^{33} + p^{51} T^{34} \) | |
| 53 | \( 1 + 366 T + 995810 T^{2} + 273030764 T^{3} + 487616941554 T^{4} + 97696592797902 T^{5} + 162874870609883027 T^{6} + 22497221882075652552 T^{7} + \)\(42\!\cdots\!63\)\( T^{8} + \)\(36\!\cdots\!26\)\( T^{9} + \)\(95\!\cdots\!32\)\( T^{10} + \)\(38\!\cdots\!36\)\( T^{11} + \)\(18\!\cdots\!06\)\( T^{12} + \)\(78\!\cdots\!90\)\( T^{13} + \)\(32\!\cdots\!41\)\( T^{14} - \)\(64\!\cdots\!92\)\( T^{15} + \)\(52\!\cdots\!54\)\( T^{16} - \)\(14\!\cdots\!88\)\( T^{17} + \)\(52\!\cdots\!54\)\( p^{3} T^{18} - \)\(64\!\cdots\!92\)\( p^{6} T^{19} + \)\(32\!\cdots\!41\)\( p^{9} T^{20} + \)\(78\!\cdots\!90\)\( p^{12} T^{21} + \)\(18\!\cdots\!06\)\( p^{15} T^{22} + \)\(38\!\cdots\!36\)\( p^{18} T^{23} + \)\(95\!\cdots\!32\)\( p^{21} T^{24} + \)\(36\!\cdots\!26\)\( p^{24} T^{25} + \)\(42\!\cdots\!63\)\( p^{27} T^{26} + 22497221882075652552 p^{30} T^{27} + 162874870609883027 p^{33} T^{28} + 97696592797902 p^{36} T^{29} + 487616941554 p^{39} T^{30} + 273030764 p^{42} T^{31} + 995810 p^{45} T^{32} + 366 p^{48} T^{33} + p^{51} T^{34} \) | |
| 59 | \( 1 + 2958 T + 6025438 T^{2} + 8881439548 T^{3} + 10870308870272 T^{4} + 11243660338431750 T^{5} + 10294357508566111571 T^{6} + \)\(84\!\cdots\!64\)\( T^{7} + \)\(63\!\cdots\!61\)\( T^{8} + \)\(43\!\cdots\!86\)\( T^{9} + \)\(28\!\cdots\!14\)\( T^{10} + \)\(17\!\cdots\!56\)\( T^{11} + \)\(99\!\cdots\!18\)\( T^{12} + \)\(54\!\cdots\!14\)\( T^{13} + \)\(28\!\cdots\!39\)\( T^{14} + \)\(14\!\cdots\!20\)\( T^{15} + \)\(69\!\cdots\!46\)\( T^{16} + \)\(32\!\cdots\!36\)\( T^{17} + \)\(69\!\cdots\!46\)\( p^{3} T^{18} + \)\(14\!\cdots\!20\)\( p^{6} T^{19} + \)\(28\!\cdots\!39\)\( p^{9} T^{20} + \)\(54\!\cdots\!14\)\( p^{12} T^{21} + \)\(99\!\cdots\!18\)\( p^{15} T^{22} + \)\(17\!\cdots\!56\)\( p^{18} T^{23} + \)\(28\!\cdots\!14\)\( p^{21} T^{24} + \)\(43\!\cdots\!86\)\( p^{24} T^{25} + \)\(63\!\cdots\!61\)\( p^{27} T^{26} + \)\(84\!\cdots\!64\)\( p^{30} T^{27} + 10294357508566111571 p^{33} T^{28} + 11243660338431750 p^{36} T^{29} + 10870308870272 p^{39} T^{30} + 8881439548 p^{42} T^{31} + 6025438 p^{45} T^{32} + 2958 p^{48} T^{33} + p^{51} T^{34} \) | |
| 61 | \( 1 + 1104 T + 2771874 T^{2} + 2397415862 T^{3} + 3457566364970 T^{4} + 2456268488256504 T^{5} + 2642560859178292383 T^{6} + \)\(15\!\cdots\!20\)\( T^{7} + \)\(14\!\cdots\!71\)\( T^{8} + \)\(72\!\cdots\!00\)\( T^{9} + \)\(57\!\cdots\!32\)\( T^{10} + \)\(25\!\cdots\!34\)\( T^{11} + \)\(18\!\cdots\!46\)\( T^{12} + \)\(72\!\cdots\!44\)\( T^{13} + \)\(50\!\cdots\!45\)\( T^{14} + \)\(18\!\cdots\!44\)\( T^{15} + \)\(12\!\cdots\!98\)\( T^{16} + \)\(41\!\cdots\!16\)\( T^{17} + \)\(12\!\cdots\!98\)\( p^{3} T^{18} + \)\(18\!\cdots\!44\)\( p^{6} T^{19} + \)\(50\!\cdots\!45\)\( p^{9} T^{20} + \)\(72\!\cdots\!44\)\( p^{12} T^{21} + \)\(18\!\cdots\!46\)\( p^{15} T^{22} + \)\(25\!\cdots\!34\)\( p^{18} T^{23} + \)\(57\!\cdots\!32\)\( p^{21} T^{24} + \)\(72\!\cdots\!00\)\( p^{24} T^{25} + \)\(14\!\cdots\!71\)\( p^{27} T^{26} + \)\(15\!\cdots\!20\)\( p^{30} T^{27} + 2642560859178292383 p^{33} T^{28} + 2456268488256504 p^{36} T^{29} + 3457566364970 p^{39} T^{30} + 2397415862 p^{42} T^{31} + 2771874 p^{45} T^{32} + 1104 p^{48} T^{33} + p^{51} T^{34} \) | |
| 67 | \( 1 - 200 T + 2068355 T^{2} - 432733944 T^{3} + 2134072505063 T^{4} - 497462439406760 T^{5} + 1481060106498590213 T^{6} - \)\(42\!\cdots\!96\)\( T^{7} + \)\(78\!\cdots\!87\)\( T^{8} - \)\(28\!\cdots\!76\)\( T^{9} + \)\(34\!\cdots\!87\)\( T^{10} - \)\(16\!\cdots\!92\)\( T^{11} + \)\(12\!\cdots\!59\)\( T^{12} - \)\(76\!\cdots\!36\)\( T^{13} + \)\(42\!\cdots\!25\)\( T^{14} - \)\(29\!\cdots\!32\)\( T^{15} + \)\(13\!\cdots\!22\)\( T^{16} - \)\(98\!\cdots\!28\)\( T^{17} + \)\(13\!\cdots\!22\)\( p^{3} T^{18} - \)\(29\!\cdots\!32\)\( p^{6} T^{19} + \)\(42\!\cdots\!25\)\( p^{9} T^{20} - \)\(76\!\cdots\!36\)\( p^{12} T^{21} + \)\(12\!\cdots\!59\)\( p^{15} T^{22} - \)\(16\!\cdots\!92\)\( p^{18} T^{23} + \)\(34\!\cdots\!87\)\( p^{21} T^{24} - \)\(28\!\cdots\!76\)\( p^{24} T^{25} + \)\(78\!\cdots\!87\)\( p^{27} T^{26} - \)\(42\!\cdots\!96\)\( p^{30} T^{27} + 1481060106498590213 p^{33} T^{28} - 497462439406760 p^{36} T^{29} + 2134072505063 p^{39} T^{30} - 432733944 p^{42} T^{31} + 2068355 p^{45} T^{32} - 200 p^{48} T^{33} + p^{51} T^{34} \) | |
| 71 | \( 1 + 3548 T + 9941717 T^{2} + 19811469728 T^{3} + 34113222645325 T^{4} + 49553600567439100 T^{5} + 64811857970672950573 T^{6} + \)\(75\!\cdots\!36\)\( T^{7} + \)\(81\!\cdots\!79\)\( T^{8} + \)\(80\!\cdots\!04\)\( T^{9} + \)\(74\!\cdots\!61\)\( T^{10} + \)\(63\!\cdots\!00\)\( T^{11} + \)\(51\!\cdots\!41\)\( T^{12} + \)\(39\!\cdots\!60\)\( T^{13} + \)\(28\!\cdots\!65\)\( T^{14} + \)\(19\!\cdots\!20\)\( T^{15} + \)\(12\!\cdots\!78\)\( T^{16} + \)\(76\!\cdots\!48\)\( T^{17} + \)\(12\!\cdots\!78\)\( p^{3} T^{18} + \)\(19\!\cdots\!20\)\( p^{6} T^{19} + \)\(28\!\cdots\!65\)\( p^{9} T^{20} + \)\(39\!\cdots\!60\)\( p^{12} T^{21} + \)\(51\!\cdots\!41\)\( p^{15} T^{22} + \)\(63\!\cdots\!00\)\( p^{18} T^{23} + \)\(74\!\cdots\!61\)\( p^{21} T^{24} + \)\(80\!\cdots\!04\)\( p^{24} T^{25} + \)\(81\!\cdots\!79\)\( p^{27} T^{26} + \)\(75\!\cdots\!36\)\( p^{30} T^{27} + 64811857970672950573 p^{33} T^{28} + 49553600567439100 p^{36} T^{29} + 34113222645325 p^{39} T^{30} + 19811469728 p^{42} T^{31} + 9941717 p^{45} T^{32} + 3548 p^{48} T^{33} + p^{51} T^{34} \) | |
| 73 | \( 1 - 1718 T + 5036921 T^{2} - 6653411452 T^{3} + 11620753427038 T^{4} - 12834152273040480 T^{5} + 17058591951347062710 T^{6} - \)\(16\!\cdots\!88\)\( T^{7} + \)\(18\!\cdots\!81\)\( T^{8} - \)\(15\!\cdots\!58\)\( T^{9} + \)\(15\!\cdots\!97\)\( T^{10} - \)\(11\!\cdots\!16\)\( T^{11} + \)\(10\!\cdots\!66\)\( T^{12} - \)\(71\!\cdots\!28\)\( T^{13} + \)\(55\!\cdots\!38\)\( T^{14} - \)\(35\!\cdots\!56\)\( T^{15} + \)\(25\!\cdots\!68\)\( T^{16} - \)\(15\!\cdots\!08\)\( T^{17} + \)\(25\!\cdots\!68\)\( p^{3} T^{18} - \)\(35\!\cdots\!56\)\( p^{6} T^{19} + \)\(55\!\cdots\!38\)\( p^{9} T^{20} - \)\(71\!\cdots\!28\)\( p^{12} T^{21} + \)\(10\!\cdots\!66\)\( p^{15} T^{22} - \)\(11\!\cdots\!16\)\( p^{18} T^{23} + \)\(15\!\cdots\!97\)\( p^{21} T^{24} - \)\(15\!\cdots\!58\)\( p^{24} T^{25} + \)\(18\!\cdots\!81\)\( p^{27} T^{26} - \)\(16\!\cdots\!88\)\( p^{30} T^{27} + 17058591951347062710 p^{33} T^{28} - 12834152273040480 p^{36} T^{29} + 11620753427038 p^{39} T^{30} - 6653411452 p^{42} T^{31} + 5036921 p^{45} T^{32} - 1718 p^{48} T^{33} + p^{51} T^{34} \) | |
| 79 | \( 1 + 352 T + 4641287 T^{2} + 1233398504 T^{3} + 10475411760466 T^{4} + 2004823685364960 T^{5} + 15568659562505427458 T^{6} + \)\(20\!\cdots\!04\)\( T^{7} + \)\(17\!\cdots\!65\)\( T^{8} + \)\(14\!\cdots\!32\)\( T^{9} + \)\(15\!\cdots\!03\)\( T^{10} + \)\(83\!\cdots\!56\)\( T^{11} + \)\(11\!\cdots\!18\)\( T^{12} + \)\(39\!\cdots\!20\)\( T^{13} + \)\(76\!\cdots\!38\)\( T^{14} + \)\(17\!\cdots\!28\)\( T^{15} + \)\(42\!\cdots\!28\)\( T^{16} + \)\(80\!\cdots\!28\)\( T^{17} + \)\(42\!\cdots\!28\)\( p^{3} T^{18} + \)\(17\!\cdots\!28\)\( p^{6} T^{19} + \)\(76\!\cdots\!38\)\( p^{9} T^{20} + \)\(39\!\cdots\!20\)\( p^{12} T^{21} + \)\(11\!\cdots\!18\)\( p^{15} T^{22} + \)\(83\!\cdots\!56\)\( p^{18} T^{23} + \)\(15\!\cdots\!03\)\( p^{21} T^{24} + \)\(14\!\cdots\!32\)\( p^{24} T^{25} + \)\(17\!\cdots\!65\)\( p^{27} T^{26} + \)\(20\!\cdots\!04\)\( p^{30} T^{27} + 15568659562505427458 p^{33} T^{28} + 2004823685364960 p^{36} T^{29} + 10475411760466 p^{39} T^{30} + 1233398504 p^{42} T^{31} + 4641287 p^{45} T^{32} + 352 p^{48} T^{33} + p^{51} T^{34} \) | |
| 83 | \( 1 + 1580 T + 4427123 T^{2} + 4618869480 T^{3} + 7888115595418 T^{4} + 5941362493790448 T^{5} + 8389038840858886402 T^{6} + \)\(47\!\cdots\!52\)\( T^{7} + \)\(69\!\cdots\!77\)\( T^{8} + \)\(33\!\cdots\!08\)\( T^{9} + \)\(55\!\cdots\!39\)\( T^{10} + \)\(25\!\cdots\!32\)\( T^{11} + \)\(42\!\cdots\!46\)\( T^{12} + \)\(18\!\cdots\!52\)\( T^{13} + \)\(29\!\cdots\!34\)\( T^{14} + \)\(11\!\cdots\!32\)\( T^{15} + \)\(17\!\cdots\!28\)\( T^{16} + \)\(64\!\cdots\!92\)\( T^{17} + \)\(17\!\cdots\!28\)\( p^{3} T^{18} + \)\(11\!\cdots\!32\)\( p^{6} T^{19} + \)\(29\!\cdots\!34\)\( p^{9} T^{20} + \)\(18\!\cdots\!52\)\( p^{12} T^{21} + \)\(42\!\cdots\!46\)\( p^{15} T^{22} + \)\(25\!\cdots\!32\)\( p^{18} T^{23} + \)\(55\!\cdots\!39\)\( p^{21} T^{24} + \)\(33\!\cdots\!08\)\( p^{24} T^{25} + \)\(69\!\cdots\!77\)\( p^{27} T^{26} + \)\(47\!\cdots\!52\)\( p^{30} T^{27} + 8389038840858886402 p^{33} T^{28} + 5941362493790448 p^{36} T^{29} + 7888115595418 p^{39} T^{30} + 4618869480 p^{42} T^{31} + 4427123 p^{45} T^{32} + 1580 p^{48} T^{33} + p^{51} T^{34} \) | |
| 89 | \( 1 + 3420 T + 11688937 T^{2} + 26416561132 T^{3} + 56871247829467 T^{4} + 100560630442209884 T^{5} + \)\(16\!\cdots\!31\)\( T^{6} + \)\(25\!\cdots\!12\)\( T^{7} + \)\(35\!\cdots\!23\)\( T^{8} + \)\(45\!\cdots\!60\)\( T^{9} + \)\(56\!\cdots\!89\)\( T^{10} + \)\(65\!\cdots\!04\)\( T^{11} + \)\(72\!\cdots\!23\)\( T^{12} + \)\(74\!\cdots\!32\)\( T^{13} + \)\(74\!\cdots\!03\)\( T^{14} + \)\(69\!\cdots\!96\)\( T^{15} + \)\(63\!\cdots\!58\)\( T^{16} + \)\(54\!\cdots\!48\)\( T^{17} + \)\(63\!\cdots\!58\)\( p^{3} T^{18} + \)\(69\!\cdots\!96\)\( p^{6} T^{19} + \)\(74\!\cdots\!03\)\( p^{9} T^{20} + \)\(74\!\cdots\!32\)\( p^{12} T^{21} + \)\(72\!\cdots\!23\)\( p^{15} T^{22} + \)\(65\!\cdots\!04\)\( p^{18} T^{23} + \)\(56\!\cdots\!89\)\( p^{21} T^{24} + \)\(45\!\cdots\!60\)\( p^{24} T^{25} + \)\(35\!\cdots\!23\)\( p^{27} T^{26} + \)\(25\!\cdots\!12\)\( p^{30} T^{27} + \)\(16\!\cdots\!31\)\( p^{33} T^{28} + 100560630442209884 p^{36} T^{29} + 56871247829467 p^{39} T^{30} + 26416561132 p^{42} T^{31} + 11688937 p^{45} T^{32} + 3420 p^{48} T^{33} + p^{51} T^{34} \) | |
| 97 | \( 1 + 2204 T + 11026092 T^{2} + 18684192630 T^{3} + 53231599573909 T^{4} + 73532631403326800 T^{5} + \)\(15\!\cdots\!37\)\( T^{6} + \)\(18\!\cdots\!06\)\( T^{7} + \)\(32\!\cdots\!08\)\( T^{8} + \)\(32\!\cdots\!36\)\( T^{9} + \)\(50\!\cdots\!13\)\( T^{10} + \)\(45\!\cdots\!52\)\( T^{11} + \)\(64\!\cdots\!18\)\( T^{12} + \)\(52\!\cdots\!12\)\( T^{13} + \)\(71\!\cdots\!12\)\( T^{14} + \)\(53\!\cdots\!72\)\( T^{15} + \)\(70\!\cdots\!10\)\( T^{16} + \)\(50\!\cdots\!20\)\( T^{17} + \)\(70\!\cdots\!10\)\( p^{3} T^{18} + \)\(53\!\cdots\!72\)\( p^{6} T^{19} + \)\(71\!\cdots\!12\)\( p^{9} T^{20} + \)\(52\!\cdots\!12\)\( p^{12} T^{21} + \)\(64\!\cdots\!18\)\( p^{15} T^{22} + \)\(45\!\cdots\!52\)\( p^{18} T^{23} + \)\(50\!\cdots\!13\)\( p^{21} T^{24} + \)\(32\!\cdots\!36\)\( p^{24} T^{25} + \)\(32\!\cdots\!08\)\( p^{27} T^{26} + \)\(18\!\cdots\!06\)\( p^{30} T^{27} + \)\(15\!\cdots\!37\)\( p^{33} T^{28} + 73532631403326800 p^{36} T^{29} + 53231599573909 p^{39} T^{30} + 18684192630 p^{42} T^{31} + 11026092 p^{45} T^{32} + 2204 p^{48} T^{33} + p^{51} T^{34} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{34} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.56096328784184429074766171392, −2.37622192554958742651215089335, −2.30154171022782196871447247211, −2.27139967423902219759088618426, −2.21261393882182627605937184606, −2.20030860216333435162538251711, −2.17331367714123129380331612115, −2.02934381500419194464685566261, −1.94009940147000720408470267610, −1.90108827010725136128054960369, −1.82512269195612688156081605763, −1.63376926310197443870420490504, −1.53160438199194411796160802810, −1.48211126947587557435066966283, −1.43181654947267117749170114550, −1.27017935432370759618001114223, −1.23065697805477872783680368355, −1.18434881522898589106415422904, −1.11489022012024518879747137689, −1.06033618406619848933997986553, −1.01852405616917994607310284277, −0.965435989741210876410261679006, −0.946265921378676131917366246313, −0.816442200721044204842009687886, −0.77795234352218720678954813687, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.77795234352218720678954813687, 0.816442200721044204842009687886, 0.946265921378676131917366246313, 0.965435989741210876410261679006, 1.01852405616917994607310284277, 1.06033618406619848933997986553, 1.11489022012024518879747137689, 1.18434881522898589106415422904, 1.23065697805477872783680368355, 1.27017935432370759618001114223, 1.43181654947267117749170114550, 1.48211126947587557435066966283, 1.53160438199194411796160802810, 1.63376926310197443870420490504, 1.82512269195612688156081605763, 1.90108827010725136128054960369, 1.94009940147000720408470267610, 2.02934381500419194464685566261, 2.17331367714123129380331612115, 2.20030860216333435162538251711, 2.21261393882182627605937184606, 2.27139967423902219759088618426, 2.30154171022782196871447247211, 2.37622192554958742651215089335, 2.56096328784184429074766171392
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.