Dirichlet series
| L(s) = 1 | + 12·3-s − 4·5-s − 105·7-s − 86·9-s − 14·11-s + 34·13-s − 48·15-s − 100·17-s + 26·19-s − 1.26e3·21-s + 158·23-s − 709·25-s − 1.56e3·27-s − 156·29-s + 252·31-s − 168·33-s + 420·35-s + 182·37-s + 408·39-s + 615·41-s + 894·43-s + 344·45-s + 1.72e3·47-s + 5.88e3·49-s − 1.20e3·51-s + 1.03e3·53-s + 56·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.357·5-s − 5.66·7-s − 3.18·9-s − 0.383·11-s + 0.725·13-s − 0.826·15-s − 1.42·17-s + 0.313·19-s − 13.0·21-s + 1.43·23-s − 5.67·25-s − 11.1·27-s − 0.998·29-s + 1.46·31-s − 0.886·33-s + 2.02·35-s + 0.808·37-s + 1.67·39-s + 2.34·41-s + 3.17·43-s + 1.13·45-s + 5.36·47-s + 17.1·49-s − 3.29·51-s + 2.67·53-s + 0.137·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{15} \cdot 41^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{15} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{15} \cdot 41^{15}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{15} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(30\) |
| Conductor: | \(2^{30} \cdot 7^{15} \cdot 41^{15}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.89814\times 10^{27}\) |
| Root analytic conductor: | \(8.23007\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((30,\ 2^{30} \cdot 7^{15} \cdot 41^{15} ,\ ( \ : [3/2]^{15} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(39.58368521\) |
| \(L(\frac12)\) | \(\approx\) | \(39.58368521\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + p T )^{15} \) | |
| 41 | \( ( 1 - p T )^{15} \) | |
| good | 3 | \( 1 - 4 p T + 230 T^{2} - 742 p T^{3} + 25066 T^{4} - 68504 p T^{5} + 583429 p T^{6} - 12520778 T^{7} + 88980338 T^{8} - 567823168 T^{9} + 3559952993 T^{10} - 6900777506 p T^{11} + 119225434088 T^{12} - 646642931044 T^{13} + 3524569153955 T^{14} - 18202489571132 T^{15} + 3524569153955 p^{3} T^{16} - 646642931044 p^{6} T^{17} + 119225434088 p^{9} T^{18} - 6900777506 p^{13} T^{19} + 3559952993 p^{15} T^{20} - 567823168 p^{18} T^{21} + 88980338 p^{21} T^{22} - 12520778 p^{24} T^{23} + 583429 p^{28} T^{24} - 68504 p^{31} T^{25} + 25066 p^{33} T^{26} - 742 p^{37} T^{27} + 230 p^{39} T^{28} - 4 p^{43} T^{29} + p^{45} T^{30} \) |
| 5 | \( 1 + 4 T + 29 p^{2} T^{2} + 3118 T^{3} + 294031 T^{4} + 320024 p T^{5} + 85452204 T^{6} + 111609968 p T^{7} + 19579957908 T^{8} + 29034513356 p T^{9} + 3747241565691 T^{10} + 5927465142066 p T^{11} + 615262950074161 T^{12} + 4906122237138696 T^{13} + 17588646723765871 p T^{14} + 672305486982180752 T^{15} + 17588646723765871 p^{4} T^{16} + 4906122237138696 p^{6} T^{17} + 615262950074161 p^{9} T^{18} + 5927465142066 p^{13} T^{19} + 3747241565691 p^{15} T^{20} + 29034513356 p^{19} T^{21} + 19579957908 p^{21} T^{22} + 111609968 p^{25} T^{23} + 85452204 p^{27} T^{24} + 320024 p^{31} T^{25} + 294031 p^{33} T^{26} + 3118 p^{36} T^{27} + 29 p^{41} T^{28} + 4 p^{42} T^{29} + p^{45} T^{30} \) | |
| 11 | \( 1 + 14 T + 9041 T^{2} + 76004 T^{3} + 41027462 T^{4} + 159404820 T^{5} + 128575741336 T^{6} + 113041363236 T^{7} + 28777059413262 p T^{8} - 261911270752352 T^{9} + 647093455403433836 T^{10} - 1142343802280652084 T^{11} + \)\(11\!\cdots\!11\)\( T^{12} - \)\(25\!\cdots\!86\)\( T^{13} + \)\(17\!\cdots\!27\)\( T^{14} - \)\(39\!\cdots\!88\)\( T^{15} + \)\(17\!\cdots\!27\)\( p^{3} T^{16} - \)\(25\!\cdots\!86\)\( p^{6} T^{17} + \)\(11\!\cdots\!11\)\( p^{9} T^{18} - 1142343802280652084 p^{12} T^{19} + 647093455403433836 p^{15} T^{20} - 261911270752352 p^{18} T^{21} + 28777059413262 p^{22} T^{22} + 113041363236 p^{24} T^{23} + 128575741336 p^{27} T^{24} + 159404820 p^{30} T^{25} + 41027462 p^{33} T^{26} + 76004 p^{36} T^{27} + 9041 p^{39} T^{28} + 14 p^{42} T^{29} + p^{45} T^{30} \) | |
| 13 | \( 1 - 34 T + 11972 T^{2} - 525042 T^{3} + 77158453 T^{4} - 3569292778 T^{5} + 353322039524 T^{6} - 15894098382726 T^{7} + 1267620683610416 T^{8} - 55206825260786088 T^{9} + 3868694714018575858 T^{10} - \)\(16\!\cdots\!46\)\( T^{11} + \)\(10\!\cdots\!11\)\( T^{12} - \)\(42\!\cdots\!12\)\( T^{13} + \)\(25\!\cdots\!21\)\( T^{14} - \)\(99\!\cdots\!64\)\( T^{15} + \)\(25\!\cdots\!21\)\( p^{3} T^{16} - \)\(42\!\cdots\!12\)\( p^{6} T^{17} + \)\(10\!\cdots\!11\)\( p^{9} T^{18} - \)\(16\!\cdots\!46\)\( p^{12} T^{19} + 3868694714018575858 p^{15} T^{20} - 55206825260786088 p^{18} T^{21} + 1267620683610416 p^{21} T^{22} - 15894098382726 p^{24} T^{23} + 353322039524 p^{27} T^{24} - 3569292778 p^{30} T^{25} + 77158453 p^{33} T^{26} - 525042 p^{36} T^{27} + 11972 p^{39} T^{28} - 34 p^{42} T^{29} + p^{45} T^{30} \) | |
| 17 | \( 1 + 100 T + 41377 T^{2} + 4048542 T^{3} + 860721239 T^{4} + 78969792706 T^{5} + 11897378405228 T^{6} + 1003181381329372 T^{7} + 122234023601828944 T^{8} + 9442788498650155842 T^{9} + \)\(99\!\cdots\!76\)\( T^{10} + \)\(70\!\cdots\!92\)\( T^{11} + \)\(66\!\cdots\!79\)\( T^{12} + \)\(44\!\cdots\!38\)\( T^{13} + \)\(38\!\cdots\!88\)\( T^{14} + \)\(23\!\cdots\!80\)\( T^{15} + \)\(38\!\cdots\!88\)\( p^{3} T^{16} + \)\(44\!\cdots\!38\)\( p^{6} T^{17} + \)\(66\!\cdots\!79\)\( p^{9} T^{18} + \)\(70\!\cdots\!92\)\( p^{12} T^{19} + \)\(99\!\cdots\!76\)\( p^{15} T^{20} + 9442788498650155842 p^{18} T^{21} + 122234023601828944 p^{21} T^{22} + 1003181381329372 p^{24} T^{23} + 11897378405228 p^{27} T^{24} + 78969792706 p^{30} T^{25} + 860721239 p^{33} T^{26} + 4048542 p^{36} T^{27} + 41377 p^{39} T^{28} + 100 p^{42} T^{29} + p^{45} T^{30} \) | |
| 19 | \( 1 - 26 T + 64675 T^{2} - 1944314 T^{3} + 2066418857 T^{4} - 68675765702 T^{5} + 43446378564535 T^{6} - 1539268768649556 T^{7} + 674296249545088226 T^{8} - 1299035156031394440 p T^{9} + \)\(82\!\cdots\!65\)\( T^{10} - \)\(30\!\cdots\!30\)\( T^{11} + \)\(81\!\cdots\!57\)\( T^{12} - \)\(28\!\cdots\!80\)\( T^{13} + \)\(66\!\cdots\!58\)\( T^{14} - \)\(22\!\cdots\!60\)\( T^{15} + \)\(66\!\cdots\!58\)\( p^{3} T^{16} - \)\(28\!\cdots\!80\)\( p^{6} T^{17} + \)\(81\!\cdots\!57\)\( p^{9} T^{18} - \)\(30\!\cdots\!30\)\( p^{12} T^{19} + \)\(82\!\cdots\!65\)\( p^{15} T^{20} - 1299035156031394440 p^{19} T^{21} + 674296249545088226 p^{21} T^{22} - 1539268768649556 p^{24} T^{23} + 43446378564535 p^{27} T^{24} - 68675765702 p^{30} T^{25} + 2066418857 p^{33} T^{26} - 1944314 p^{36} T^{27} + 64675 p^{39} T^{28} - 26 p^{42} T^{29} + p^{45} T^{30} \) | |
| 23 | \( 1 - 158 T + 108109 T^{2} - 627600 p T^{3} + 5726295399 T^{4} - 667651571082 T^{5} + 200325913647651 T^{6} - 20845285472253364 T^{7} + 5231935898529502040 T^{8} - \)\(49\!\cdots\!32\)\( T^{9} + \)\(10\!\cdots\!83\)\( T^{10} - \)\(93\!\cdots\!66\)\( T^{11} + \)\(18\!\cdots\!23\)\( T^{12} - \)\(14\!\cdots\!64\)\( T^{13} + \)\(26\!\cdots\!52\)\( T^{14} - \)\(19\!\cdots\!84\)\( T^{15} + \)\(26\!\cdots\!52\)\( p^{3} T^{16} - \)\(14\!\cdots\!64\)\( p^{6} T^{17} + \)\(18\!\cdots\!23\)\( p^{9} T^{18} - \)\(93\!\cdots\!66\)\( p^{12} T^{19} + \)\(10\!\cdots\!83\)\( p^{15} T^{20} - \)\(49\!\cdots\!32\)\( p^{18} T^{21} + 5231935898529502040 p^{21} T^{22} - 20845285472253364 p^{24} T^{23} + 200325913647651 p^{27} T^{24} - 667651571082 p^{30} T^{25} + 5726295399 p^{33} T^{26} - 627600 p^{37} T^{27} + 108109 p^{39} T^{28} - 158 p^{42} T^{29} + p^{45} T^{30} \) | |
| 29 | \( 1 + 156 T + 156081 T^{2} + 28089984 T^{3} + 13710829511 T^{4} + 2387339575872 T^{5} + 848693884036442 T^{6} + 137711651602491028 T^{7} + 40203685256803832974 T^{8} + \)\(60\!\cdots\!28\)\( T^{9} + \)\(15\!\cdots\!27\)\( T^{10} + \)\(21\!\cdots\!00\)\( T^{11} + \)\(50\!\cdots\!81\)\( T^{12} + \)\(66\!\cdots\!08\)\( T^{13} + \)\(14\!\cdots\!91\)\( T^{14} + \)\(17\!\cdots\!56\)\( T^{15} + \)\(14\!\cdots\!91\)\( p^{3} T^{16} + \)\(66\!\cdots\!08\)\( p^{6} T^{17} + \)\(50\!\cdots\!81\)\( p^{9} T^{18} + \)\(21\!\cdots\!00\)\( p^{12} T^{19} + \)\(15\!\cdots\!27\)\( p^{15} T^{20} + \)\(60\!\cdots\!28\)\( p^{18} T^{21} + 40203685256803832974 p^{21} T^{22} + 137711651602491028 p^{24} T^{23} + 848693884036442 p^{27} T^{24} + 2387339575872 p^{30} T^{25} + 13710829511 p^{33} T^{26} + 28089984 p^{36} T^{27} + 156081 p^{39} T^{28} + 156 p^{42} T^{29} + p^{45} T^{30} \) | |
| 31 | \( 1 - 252 T + 219123 T^{2} - 44955746 T^{3} + 22783413941 T^{4} - 4093435264170 T^{5} + 1566441430304816 T^{6} - 259734239399285314 T^{7} + 82259641841244087602 T^{8} - \)\(13\!\cdots\!86\)\( T^{9} + \)\(35\!\cdots\!29\)\( T^{10} - \)\(55\!\cdots\!30\)\( T^{11} + \)\(13\!\cdots\!35\)\( T^{12} - \)\(20\!\cdots\!64\)\( T^{13} + \)\(44\!\cdots\!07\)\( T^{14} - \)\(64\!\cdots\!92\)\( T^{15} + \)\(44\!\cdots\!07\)\( p^{3} T^{16} - \)\(20\!\cdots\!64\)\( p^{6} T^{17} + \)\(13\!\cdots\!35\)\( p^{9} T^{18} - \)\(55\!\cdots\!30\)\( p^{12} T^{19} + \)\(35\!\cdots\!29\)\( p^{15} T^{20} - \)\(13\!\cdots\!86\)\( p^{18} T^{21} + 82259641841244087602 p^{21} T^{22} - 259734239399285314 p^{24} T^{23} + 1566441430304816 p^{27} T^{24} - 4093435264170 p^{30} T^{25} + 22783413941 p^{33} T^{26} - 44955746 p^{36} T^{27} + 219123 p^{39} T^{28} - 252 p^{42} T^{29} + p^{45} T^{30} \) | |
| 37 | \( 1 - 182 T + 285164 T^{2} - 33188018 T^{3} + 41062706672 T^{4} - 2617336683446 T^{5} + 4132839894287892 T^{6} - 82942579272773500 T^{7} + \)\(33\!\cdots\!74\)\( T^{8} + \)\(62\!\cdots\!82\)\( T^{9} + \)\(22\!\cdots\!52\)\( T^{10} + \)\(11\!\cdots\!18\)\( T^{11} + \)\(13\!\cdots\!44\)\( T^{12} + \)\(97\!\cdots\!10\)\( T^{13} + \)\(72\!\cdots\!29\)\( T^{14} + \)\(57\!\cdots\!72\)\( T^{15} + \)\(72\!\cdots\!29\)\( p^{3} T^{16} + \)\(97\!\cdots\!10\)\( p^{6} T^{17} + \)\(13\!\cdots\!44\)\( p^{9} T^{18} + \)\(11\!\cdots\!18\)\( p^{12} T^{19} + \)\(22\!\cdots\!52\)\( p^{15} T^{20} + \)\(62\!\cdots\!82\)\( p^{18} T^{21} + \)\(33\!\cdots\!74\)\( p^{21} T^{22} - 82942579272773500 p^{24} T^{23} + 4132839894287892 p^{27} T^{24} - 2617336683446 p^{30} T^{25} + 41062706672 p^{33} T^{26} - 33188018 p^{36} T^{27} + 285164 p^{39} T^{28} - 182 p^{42} T^{29} + p^{45} T^{30} \) | |
| 43 | \( 1 - 894 T + 1010247 T^{2} - 656660318 T^{3} + 457541335801 T^{4} - 240409772342680 T^{5} + 129238128451836290 T^{6} - 57802342571803600884 T^{7} + \)\(25\!\cdots\!14\)\( T^{8} - \)\(10\!\cdots\!52\)\( T^{9} + \)\(39\!\cdots\!86\)\( T^{10} - \)\(13\!\cdots\!52\)\( T^{11} + \)\(48\!\cdots\!05\)\( T^{12} - \)\(15\!\cdots\!18\)\( T^{13} + \)\(46\!\cdots\!10\)\( T^{14} - \)\(13\!\cdots\!92\)\( T^{15} + \)\(46\!\cdots\!10\)\( p^{3} T^{16} - \)\(15\!\cdots\!18\)\( p^{6} T^{17} + \)\(48\!\cdots\!05\)\( p^{9} T^{18} - \)\(13\!\cdots\!52\)\( p^{12} T^{19} + \)\(39\!\cdots\!86\)\( p^{15} T^{20} - \)\(10\!\cdots\!52\)\( p^{18} T^{21} + \)\(25\!\cdots\!14\)\( p^{21} T^{22} - 57802342571803600884 p^{24} T^{23} + 129238128451836290 p^{27} T^{24} - 240409772342680 p^{30} T^{25} + 457541335801 p^{33} T^{26} - 656660318 p^{36} T^{27} + 1010247 p^{39} T^{28} - 894 p^{42} T^{29} + p^{45} T^{30} \) | |
| 47 | \( 1 - 1728 T + 2000486 T^{2} - 1706082616 T^{3} + 1226204802536 T^{4} - 757498217198562 T^{5} + 420605011864799588 T^{6} - \)\(21\!\cdots\!48\)\( T^{7} + \)\(98\!\cdots\!90\)\( T^{8} - \)\(42\!\cdots\!70\)\( T^{9} + \)\(17\!\cdots\!60\)\( T^{10} - \)\(67\!\cdots\!60\)\( T^{11} + \)\(24\!\cdots\!18\)\( T^{12} - \)\(18\!\cdots\!12\)\( p T^{13} + \)\(30\!\cdots\!31\)\( T^{14} - \)\(98\!\cdots\!16\)\( T^{15} + \)\(30\!\cdots\!31\)\( p^{3} T^{16} - \)\(18\!\cdots\!12\)\( p^{7} T^{17} + \)\(24\!\cdots\!18\)\( p^{9} T^{18} - \)\(67\!\cdots\!60\)\( p^{12} T^{19} + \)\(17\!\cdots\!60\)\( p^{15} T^{20} - \)\(42\!\cdots\!70\)\( p^{18} T^{21} + \)\(98\!\cdots\!90\)\( p^{21} T^{22} - \)\(21\!\cdots\!48\)\( p^{24} T^{23} + 420605011864799588 p^{27} T^{24} - 757498217198562 p^{30} T^{25} + 1226204802536 p^{33} T^{26} - 1706082616 p^{36} T^{27} + 2000486 p^{39} T^{28} - 1728 p^{42} T^{29} + p^{45} T^{30} \) | |
| 53 | \( 1 - 1034 T + 1826561 T^{2} - 1466506492 T^{3} + 1495161568831 T^{4} - 992462720221790 T^{5} + 753479581604360646 T^{6} - \)\(42\!\cdots\!62\)\( T^{7} + \)\(26\!\cdots\!02\)\( T^{8} - \)\(13\!\cdots\!66\)\( T^{9} + \)\(72\!\cdots\!71\)\( T^{10} - \)\(32\!\cdots\!84\)\( T^{11} + \)\(15\!\cdots\!09\)\( T^{12} - \)\(63\!\cdots\!30\)\( T^{13} + \)\(27\!\cdots\!55\)\( T^{14} - \)\(10\!\cdots\!04\)\( T^{15} + \)\(27\!\cdots\!55\)\( p^{3} T^{16} - \)\(63\!\cdots\!30\)\( p^{6} T^{17} + \)\(15\!\cdots\!09\)\( p^{9} T^{18} - \)\(32\!\cdots\!84\)\( p^{12} T^{19} + \)\(72\!\cdots\!71\)\( p^{15} T^{20} - \)\(13\!\cdots\!66\)\( p^{18} T^{21} + \)\(26\!\cdots\!02\)\( p^{21} T^{22} - \)\(42\!\cdots\!62\)\( p^{24} T^{23} + 753479581604360646 p^{27} T^{24} - 992462720221790 p^{30} T^{25} + 1495161568831 p^{33} T^{26} - 1466506492 p^{36} T^{27} + 1826561 p^{39} T^{28} - 1034 p^{42} T^{29} + p^{45} T^{30} \) | |
| 59 | \( 1 - 262 T + 2304264 T^{2} - 648137094 T^{3} + 2583106499953 T^{4} - 753697626008440 T^{5} + 1869461363119853360 T^{6} - \)\(54\!\cdots\!76\)\( T^{7} + \)\(97\!\cdots\!46\)\( T^{8} - \)\(28\!\cdots\!28\)\( T^{9} + \)\(39\!\cdots\!63\)\( T^{10} - \)\(10\!\cdots\!58\)\( T^{11} + \)\(12\!\cdots\!18\)\( T^{12} - \)\(31\!\cdots\!74\)\( T^{13} + \)\(31\!\cdots\!59\)\( T^{14} - \)\(73\!\cdots\!12\)\( T^{15} + \)\(31\!\cdots\!59\)\( p^{3} T^{16} - \)\(31\!\cdots\!74\)\( p^{6} T^{17} + \)\(12\!\cdots\!18\)\( p^{9} T^{18} - \)\(10\!\cdots\!58\)\( p^{12} T^{19} + \)\(39\!\cdots\!63\)\( p^{15} T^{20} - \)\(28\!\cdots\!28\)\( p^{18} T^{21} + \)\(97\!\cdots\!46\)\( p^{21} T^{22} - \)\(54\!\cdots\!76\)\( p^{24} T^{23} + 1869461363119853360 p^{27} T^{24} - 753697626008440 p^{30} T^{25} + 2583106499953 p^{33} T^{26} - 648137094 p^{36} T^{27} + 2304264 p^{39} T^{28} - 262 p^{42} T^{29} + p^{45} T^{30} \) | |
| 61 | \( 1 - 322 T + 2032688 T^{2} - 651939460 T^{3} + 2043441787843 T^{4} - 650914658380300 T^{5} + 1350947784921717983 T^{6} - \)\(42\!\cdots\!56\)\( T^{7} + \)\(66\!\cdots\!67\)\( T^{8} - \)\(20\!\cdots\!72\)\( T^{9} + \)\(25\!\cdots\!49\)\( T^{10} - \)\(79\!\cdots\!56\)\( T^{11} + \)\(80\!\cdots\!80\)\( T^{12} - \)\(24\!\cdots\!62\)\( T^{13} + \)\(21\!\cdots\!17\)\( T^{14} - \)\(60\!\cdots\!32\)\( T^{15} + \)\(21\!\cdots\!17\)\( p^{3} T^{16} - \)\(24\!\cdots\!62\)\( p^{6} T^{17} + \)\(80\!\cdots\!80\)\( p^{9} T^{18} - \)\(79\!\cdots\!56\)\( p^{12} T^{19} + \)\(25\!\cdots\!49\)\( p^{15} T^{20} - \)\(20\!\cdots\!72\)\( p^{18} T^{21} + \)\(66\!\cdots\!67\)\( p^{21} T^{22} - \)\(42\!\cdots\!56\)\( p^{24} T^{23} + 1350947784921717983 p^{27} T^{24} - 650914658380300 p^{30} T^{25} + 2043441787843 p^{33} T^{26} - 651939460 p^{36} T^{27} + 2032688 p^{39} T^{28} - 322 p^{42} T^{29} + p^{45} T^{30} \) | |
| 67 | \( 1 - 1808 T + 3162662 T^{2} - 3556350012 T^{3} + 3726043683411 T^{4} - 3170706451613926 T^{5} + 2535123489209898932 T^{6} - \)\(18\!\cdots\!64\)\( T^{7} + \)\(12\!\cdots\!74\)\( T^{8} - \)\(82\!\cdots\!54\)\( T^{9} + \)\(52\!\cdots\!57\)\( T^{10} - \)\(32\!\cdots\!00\)\( T^{11} + \)\(20\!\cdots\!52\)\( T^{12} - \)\(11\!\cdots\!92\)\( T^{13} + \)\(67\!\cdots\!95\)\( T^{14} - \)\(37\!\cdots\!52\)\( T^{15} + \)\(67\!\cdots\!95\)\( p^{3} T^{16} - \)\(11\!\cdots\!92\)\( p^{6} T^{17} + \)\(20\!\cdots\!52\)\( p^{9} T^{18} - \)\(32\!\cdots\!00\)\( p^{12} T^{19} + \)\(52\!\cdots\!57\)\( p^{15} T^{20} - \)\(82\!\cdots\!54\)\( p^{18} T^{21} + \)\(12\!\cdots\!74\)\( p^{21} T^{22} - \)\(18\!\cdots\!64\)\( p^{24} T^{23} + 2535123489209898932 p^{27} T^{24} - 3170706451613926 p^{30} T^{25} + 3726043683411 p^{33} T^{26} - 3556350012 p^{36} T^{27} + 3162662 p^{39} T^{28} - 1808 p^{42} T^{29} + p^{45} T^{30} \) | |
| 71 | \( 1 - 584 T + 2925078 T^{2} - 2059117128 T^{3} + 4313602924011 T^{4} - 3313563734077576 T^{5} + 4305802493435127997 T^{6} - \)\(33\!\cdots\!40\)\( T^{7} + \)\(32\!\cdots\!77\)\( T^{8} - \)\(24\!\cdots\!08\)\( T^{9} + \)\(19\!\cdots\!81\)\( T^{10} - \)\(13\!\cdots\!32\)\( T^{11} + \)\(97\!\cdots\!64\)\( T^{12} - \)\(61\!\cdots\!16\)\( T^{13} + \)\(40\!\cdots\!65\)\( T^{14} - \)\(23\!\cdots\!52\)\( T^{15} + \)\(40\!\cdots\!65\)\( p^{3} T^{16} - \)\(61\!\cdots\!16\)\( p^{6} T^{17} + \)\(97\!\cdots\!64\)\( p^{9} T^{18} - \)\(13\!\cdots\!32\)\( p^{12} T^{19} + \)\(19\!\cdots\!81\)\( p^{15} T^{20} - \)\(24\!\cdots\!08\)\( p^{18} T^{21} + \)\(32\!\cdots\!77\)\( p^{21} T^{22} - \)\(33\!\cdots\!40\)\( p^{24} T^{23} + 4305802493435127997 p^{27} T^{24} - 3313563734077576 p^{30} T^{25} + 4313602924011 p^{33} T^{26} - 2059117128 p^{36} T^{27} + 2925078 p^{39} T^{28} - 584 p^{42} T^{29} + p^{45} T^{30} \) | |
| 73 | \( 1 + 1290 T + 2802525 T^{2} + 2714753908 T^{3} + 3598469629664 T^{4} + 3050674908588252 T^{5} + 3190006345863700182 T^{6} + \)\(25\!\cdots\!08\)\( T^{7} + \)\(22\!\cdots\!00\)\( T^{8} + \)\(16\!\cdots\!88\)\( T^{9} + \)\(13\!\cdots\!86\)\( T^{10} + \)\(95\!\cdots\!48\)\( T^{11} + \)\(68\!\cdots\!09\)\( T^{12} + \)\(45\!\cdots\!02\)\( T^{13} + \)\(30\!\cdots\!25\)\( T^{14} + \)\(19\!\cdots\!56\)\( T^{15} + \)\(30\!\cdots\!25\)\( p^{3} T^{16} + \)\(45\!\cdots\!02\)\( p^{6} T^{17} + \)\(68\!\cdots\!09\)\( p^{9} T^{18} + \)\(95\!\cdots\!48\)\( p^{12} T^{19} + \)\(13\!\cdots\!86\)\( p^{15} T^{20} + \)\(16\!\cdots\!88\)\( p^{18} T^{21} + \)\(22\!\cdots\!00\)\( p^{21} T^{22} + \)\(25\!\cdots\!08\)\( p^{24} T^{23} + 3190006345863700182 p^{27} T^{24} + 3050674908588252 p^{30} T^{25} + 3598469629664 p^{33} T^{26} + 2714753908 p^{36} T^{27} + 2802525 p^{39} T^{28} + 1290 p^{42} T^{29} + p^{45} T^{30} \) | |
| 79 | \( 1 - 3726 T + 10277740 T^{2} - 20572649318 T^{3} + 35281267790502 T^{4} - 51646045478690226 T^{5} + 68189253395606600863 T^{6} - \)\(81\!\cdots\!12\)\( T^{7} + \)\(89\!\cdots\!79\)\( T^{8} - \)\(90\!\cdots\!42\)\( T^{9} + \)\(86\!\cdots\!18\)\( T^{10} - \)\(77\!\cdots\!82\)\( T^{11} + \)\(65\!\cdots\!40\)\( T^{12} - \)\(52\!\cdots\!06\)\( T^{13} + \)\(39\!\cdots\!37\)\( T^{14} - \)\(28\!\cdots\!68\)\( T^{15} + \)\(39\!\cdots\!37\)\( p^{3} T^{16} - \)\(52\!\cdots\!06\)\( p^{6} T^{17} + \)\(65\!\cdots\!40\)\( p^{9} T^{18} - \)\(77\!\cdots\!82\)\( p^{12} T^{19} + \)\(86\!\cdots\!18\)\( p^{15} T^{20} - \)\(90\!\cdots\!42\)\( p^{18} T^{21} + \)\(89\!\cdots\!79\)\( p^{21} T^{22} - \)\(81\!\cdots\!12\)\( p^{24} T^{23} + 68189253395606600863 p^{27} T^{24} - 51646045478690226 p^{30} T^{25} + 35281267790502 p^{33} T^{26} - 20572649318 p^{36} T^{27} + 10277740 p^{39} T^{28} - 3726 p^{42} T^{29} + p^{45} T^{30} \) | |
| 83 | \( 1 - 2484 T + 6916465 T^{2} - 10316336740 T^{3} + 16623120355738 T^{4} - 17646202986806120 T^{5} + 21219015992629694560 T^{6} - \)\(17\!\cdots\!20\)\( T^{7} + \)\(17\!\cdots\!26\)\( T^{8} - \)\(11\!\cdots\!72\)\( T^{9} + \)\(10\!\cdots\!88\)\( T^{10} - \)\(48\!\cdots\!28\)\( T^{11} + \)\(49\!\cdots\!27\)\( T^{12} - \)\(13\!\cdots\!04\)\( T^{13} + \)\(20\!\cdots\!19\)\( T^{14} - \)\(38\!\cdots\!16\)\( T^{15} + \)\(20\!\cdots\!19\)\( p^{3} T^{16} - \)\(13\!\cdots\!04\)\( p^{6} T^{17} + \)\(49\!\cdots\!27\)\( p^{9} T^{18} - \)\(48\!\cdots\!28\)\( p^{12} T^{19} + \)\(10\!\cdots\!88\)\( p^{15} T^{20} - \)\(11\!\cdots\!72\)\( p^{18} T^{21} + \)\(17\!\cdots\!26\)\( p^{21} T^{22} - \)\(17\!\cdots\!20\)\( p^{24} T^{23} + 21219015992629694560 p^{27} T^{24} - 17646202986806120 p^{30} T^{25} + 16623120355738 p^{33} T^{26} - 10316336740 p^{36} T^{27} + 6916465 p^{39} T^{28} - 2484 p^{42} T^{29} + p^{45} T^{30} \) | |
| 89 | \( 1 - 876 T + 6440938 T^{2} - 6182974200 T^{3} + 21317711959016 T^{4} - 21298058259902854 T^{5} + 47540955679244200795 T^{6} - \)\(47\!\cdots\!50\)\( T^{7} + \)\(79\!\cdots\!06\)\( T^{8} - \)\(76\!\cdots\!30\)\( T^{9} + \)\(10\!\cdots\!63\)\( T^{10} - \)\(94\!\cdots\!82\)\( T^{11} + \)\(10\!\cdots\!04\)\( T^{12} - \)\(92\!\cdots\!66\)\( T^{13} + \)\(93\!\cdots\!89\)\( T^{14} - \)\(72\!\cdots\!68\)\( T^{15} + \)\(93\!\cdots\!89\)\( p^{3} T^{16} - \)\(92\!\cdots\!66\)\( p^{6} T^{17} + \)\(10\!\cdots\!04\)\( p^{9} T^{18} - \)\(94\!\cdots\!82\)\( p^{12} T^{19} + \)\(10\!\cdots\!63\)\( p^{15} T^{20} - \)\(76\!\cdots\!30\)\( p^{18} T^{21} + \)\(79\!\cdots\!06\)\( p^{21} T^{22} - \)\(47\!\cdots\!50\)\( p^{24} T^{23} + 47540955679244200795 p^{27} T^{24} - 21298058259902854 p^{30} T^{25} + 21317711959016 p^{33} T^{26} - 6182974200 p^{36} T^{27} + 6440938 p^{39} T^{28} - 876 p^{42} T^{29} + p^{45} T^{30} \) | |
| 97 | \( 1 + 154 T + 7230468 T^{2} + 2830257360 T^{3} + 26076659055728 T^{4} + 15851847976485136 T^{5} + 63970924682101120125 T^{6} + \)\(49\!\cdots\!88\)\( T^{7} + \)\(12\!\cdots\!46\)\( T^{8} + \)\(10\!\cdots\!08\)\( T^{9} + \)\(18\!\cdots\!71\)\( T^{10} + \)\(16\!\cdots\!42\)\( T^{11} + \)\(24\!\cdots\!70\)\( T^{12} + \)\(19\!\cdots\!36\)\( T^{13} + \)\(26\!\cdots\!15\)\( T^{14} + \)\(19\!\cdots\!72\)\( T^{15} + \)\(26\!\cdots\!15\)\( p^{3} T^{16} + \)\(19\!\cdots\!36\)\( p^{6} T^{17} + \)\(24\!\cdots\!70\)\( p^{9} T^{18} + \)\(16\!\cdots\!42\)\( p^{12} T^{19} + \)\(18\!\cdots\!71\)\( p^{15} T^{20} + \)\(10\!\cdots\!08\)\( p^{18} T^{21} + \)\(12\!\cdots\!46\)\( p^{21} T^{22} + \)\(49\!\cdots\!88\)\( p^{24} T^{23} + 63970924682101120125 p^{27} T^{24} + 15851847976485136 p^{30} T^{25} + 26076659055728 p^{33} T^{26} + 2830257360 p^{36} T^{27} + 7230468 p^{39} T^{28} + 154 p^{42} T^{29} + p^{45} T^{30} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{30} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.24126367313056878093045136618, −2.22261213235441130699811470627, −2.20810801527206915537135551564, −2.19004685241227518095354080797, −2.05868310762276392839584545074, −2.04345652269163774859222155565, −1.95397353833904913539941957124, −1.93826003329298206748803198676, −1.69008613058503306243357331621, −1.54920467043184849625759591045, −1.27251770940532284907386113469, −1.09119721153263340962028168848, −0.987472007169017908480951112315, −0.891265930700902874368156434305, −0.820844362122866540675766380037, −0.78589329047755835944327830549, −0.69981395771646563331595594602, −0.64477153681187387352922187256, −0.59862243715772053646469106110, −0.56615615398648534711352526388, −0.48380138140311619651795653915, −0.44144433384927007377797886024, −0.32516486896279506641701102716, −0.21794793318863166498401064941, −0.15694541762608386239012946009, 0.15694541762608386239012946009, 0.21794793318863166498401064941, 0.32516486896279506641701102716, 0.44144433384927007377797886024, 0.48380138140311619651795653915, 0.56615615398648534711352526388, 0.59862243715772053646469106110, 0.64477153681187387352922187256, 0.69981395771646563331595594602, 0.78589329047755835944327830549, 0.820844362122866540675766380037, 0.891265930700902874368156434305, 0.987472007169017908480951112315, 1.09119721153263340962028168848, 1.27251770940532284907386113469, 1.54920467043184849625759591045, 1.69008613058503306243357331621, 1.93826003329298206748803198676, 1.95397353833904913539941957124, 2.04345652269163774859222155565, 2.05868310762276392839584545074, 2.19004685241227518095354080797, 2.20810801527206915537135551564, 2.22261213235441130699811470627, 2.24126367313056878093045136618
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.