Dirichlet series
| L(s) = 1 | + 30·5-s − 12·7-s + 84·13-s + 24·17-s − 228·19-s − 30·23-s + 375·25-s − 168·29-s + 324·31-s − 360·35-s − 984·37-s − 312·41-s + 156·43-s − 462·47-s + 1.39e3·49-s + 2.02e3·53-s − 1.00e3·59-s − 36·61-s + 2.52e3·65-s − 144·67-s + 2.42e3·71-s − 1.80e3·73-s + 936·79-s − 288·83-s + 720·85-s − 240·89-s − 1.00e3·91-s + ⋯ |
| L(s) = 1 | + 2.68·5-s − 0.647·7-s + 1.79·13-s + 0.342·17-s − 2.75·19-s − 0.271·23-s + 3·25-s − 1.07·29-s + 1.87·31-s − 1.73·35-s − 4.37·37-s − 1.18·41-s + 0.553·43-s − 1.43·47-s + 4.06·49-s + 5.25·53-s − 2.22·59-s − 0.0755·61-s + 4.80·65-s − 0.262·67-s + 4.05·71-s − 2.88·73-s + 1.33·79-s − 0.380·83-s + 0.918·85-s − 0.285·89-s − 1.16·91-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{48} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{48} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 3^{48} \cdot 5^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.81531\times 10^{23}\) |
| Root analytic conductor: | \(9.77666\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 3^{48} \cdot 5^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.8005808408\) |
| \(L(\frac12)\) | \(\approx\) | \(0.8005808408\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 5 | \( ( 1 - p T + p^{2} T^{2} )^{6} \) | |
| good | 7 | \( 1 + 12 T - 1251 T^{2} - 8852 T^{3} + 869706 T^{4} + 2418636 T^{5} - 432468305 T^{6} + 6818796 T^{7} + 174079179000 T^{8} - 27838473596 p T^{9} - 63016908924507 T^{10} + 6168948456420 p T^{11} + 22062971352343132 T^{12} + 6168948456420 p^{4} T^{13} - 63016908924507 p^{6} T^{14} - 27838473596 p^{10} T^{15} + 174079179000 p^{12} T^{16} + 6818796 p^{15} T^{17} - 432468305 p^{18} T^{18} + 2418636 p^{21} T^{19} + 869706 p^{24} T^{20} - 8852 p^{27} T^{21} - 1251 p^{30} T^{22} + 12 p^{33} T^{23} + p^{36} T^{24} \) |
| 11 | \( 1 - 2160 T^{2} + 120960 T^{3} + 2409840 T^{4} - 303004800 T^{5} + 1802722964 T^{6} + 378779587200 T^{7} - 12591300800880 T^{8} - 416666222017920 T^{9} + 22916755352225520 T^{10} + 225269236344902400 T^{11} - 32556278352741660714 T^{12} + 225269236344902400 p^{3} T^{13} + 22916755352225520 p^{6} T^{14} - 416666222017920 p^{9} T^{15} - 12591300800880 p^{12} T^{16} + 378779587200 p^{15} T^{17} + 1802722964 p^{18} T^{18} - 303004800 p^{21} T^{19} + 2409840 p^{24} T^{20} + 120960 p^{27} T^{21} - 2160 p^{30} T^{22} + p^{36} T^{24} \) | |
| 13 | \( 1 - 84 T - 1191 T^{2} + 390892 T^{3} - 12078858 T^{4} - 573946284 T^{5} + 39760741927 T^{6} + 567861605028 T^{7} - 94514947434720 T^{8} + 707801909639308 T^{9} + 142145569197900069 T^{10} - 224925417480214884 p T^{11} - 504261087884314700 p^{2} T^{12} - 224925417480214884 p^{4} T^{13} + 142145569197900069 p^{6} T^{14} + 707801909639308 p^{9} T^{15} - 94514947434720 p^{12} T^{16} + 567861605028 p^{15} T^{17} + 39760741927 p^{18} T^{18} - 573946284 p^{21} T^{19} - 12078858 p^{24} T^{20} + 390892 p^{27} T^{21} - 1191 p^{30} T^{22} - 84 p^{33} T^{23} + p^{36} T^{24} \) | |
| 17 | \( ( 1 - 12 T + 14850 T^{2} + 129084 T^{3} + 118074771 T^{4} + 1314249936 T^{5} + 701082237940 T^{6} + 1314249936 p^{3} T^{7} + 118074771 p^{6} T^{8} + 129084 p^{9} T^{9} + 14850 p^{12} T^{10} - 12 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 19 | \( ( 1 + 6 p T + 17145 T^{2} + 1907786 T^{3} + 213615402 T^{4} + 17972482722 T^{5} + 1791165486729 T^{6} + 17972482722 p^{3} T^{7} + 213615402 p^{6} T^{8} + 1907786 p^{9} T^{9} + 17145 p^{12} T^{10} + 6 p^{16} T^{11} + p^{18} T^{12} )^{2} \) | |
| 23 | \( 1 + 30 T - 32067 T^{2} - 1003650 T^{3} + 399772098 T^{4} + 8349369870 T^{5} - 2941966234309 T^{6} + 124515107215110 T^{7} + 30963714482206584 T^{8} - 2978669391446443050 T^{9} - \)\(29\!\cdots\!95\)\( T^{10} + \)\(17\!\cdots\!90\)\( T^{11} + \)\(19\!\cdots\!96\)\( T^{12} + \)\(17\!\cdots\!90\)\( p^{3} T^{13} - \)\(29\!\cdots\!95\)\( p^{6} T^{14} - 2978669391446443050 p^{9} T^{15} + 30963714482206584 p^{12} T^{16} + 124515107215110 p^{15} T^{17} - 2941966234309 p^{18} T^{18} + 8349369870 p^{21} T^{19} + 399772098 p^{24} T^{20} - 1003650 p^{27} T^{21} - 32067 p^{30} T^{22} + 30 p^{33} T^{23} + p^{36} T^{24} \) | |
| 29 | \( 1 + 168 T - 936 p T^{2} + 2146848 T^{3} + 2345376024 T^{4} - 67036302264 T^{5} - 17841214454764 T^{6} + 14664211263213048 T^{7} + 616365708406453512 T^{8} - \)\(20\!\cdots\!84\)\( T^{9} + \)\(53\!\cdots\!28\)\( T^{10} + \)\(62\!\cdots\!60\)\( T^{11} - \)\(82\!\cdots\!54\)\( T^{12} + \)\(62\!\cdots\!60\)\( p^{3} T^{13} + \)\(53\!\cdots\!28\)\( p^{6} T^{14} - \)\(20\!\cdots\!84\)\( p^{9} T^{15} + 616365708406453512 p^{12} T^{16} + 14664211263213048 p^{15} T^{17} - 17841214454764 p^{18} T^{18} - 67036302264 p^{21} T^{19} + 2345376024 p^{24} T^{20} + 2146848 p^{27} T^{21} - 936 p^{31} T^{22} + 168 p^{33} T^{23} + p^{36} T^{24} \) | |
| 31 | \( 1 - 324 T + 7176 T^{2} - 2173928 T^{3} + 2333378208 T^{4} - 43640086500 T^{5} - 7135168403060 T^{6} - 17215479605560212 T^{7} + 981906575578347024 T^{8} + \)\(18\!\cdots\!48\)\( T^{9} + \)\(81\!\cdots\!48\)\( T^{10} - \)\(75\!\cdots\!48\)\( T^{11} - \)\(20\!\cdots\!42\)\( T^{12} - \)\(75\!\cdots\!48\)\( p^{3} T^{13} + \)\(81\!\cdots\!48\)\( p^{6} T^{14} + \)\(18\!\cdots\!48\)\( p^{9} T^{15} + 981906575578347024 p^{12} T^{16} - 17215479605560212 p^{15} T^{17} - 7135168403060 p^{18} T^{18} - 43640086500 p^{21} T^{19} + 2333378208 p^{24} T^{20} - 2173928 p^{27} T^{21} + 7176 p^{30} T^{22} - 324 p^{33} T^{23} + p^{36} T^{24} \) | |
| 37 | \( ( 1 + 492 T + 243462 T^{2} + 63512132 T^{3} + 20625037659 T^{4} + 4455049435488 T^{5} + 1247627202498492 T^{6} + 4455049435488 p^{3} T^{7} + 20625037659 p^{6} T^{8} + 63512132 p^{9} T^{9} + 243462 p^{12} T^{10} + 492 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 41 | \( 1 + 312 T - 120621 T^{2} - 45815160 T^{3} + 5043089523 T^{4} + 2200268726304 T^{5} + 241127384817236 T^{6} + 118759958219866032 T^{7} - 16893543217280182695 T^{8} - \)\(21\!\cdots\!52\)\( T^{9} - \)\(24\!\cdots\!91\)\( T^{10} + \)\(86\!\cdots\!44\)\( T^{11} + \)\(36\!\cdots\!50\)\( T^{12} + \)\(86\!\cdots\!44\)\( p^{3} T^{13} - \)\(24\!\cdots\!91\)\( p^{6} T^{14} - \)\(21\!\cdots\!52\)\( p^{9} T^{15} - 16893543217280182695 p^{12} T^{16} + 118759958219866032 p^{15} T^{17} + 241127384817236 p^{18} T^{18} + 2200268726304 p^{21} T^{19} + 5043089523 p^{24} T^{20} - 45815160 p^{27} T^{21} - 120621 p^{30} T^{22} + 312 p^{33} T^{23} + p^{36} T^{24} \) | |
| 43 | \( 1 - 156 T - 194538 T^{2} - 11259248 T^{3} + 17587396185 T^{4} + 4649175328776 T^{5} - 585614628341414 T^{6} - 152529932331002772 T^{7} + 22258041301325289726 T^{8} - \)\(30\!\cdots\!76\)\( T^{9} - \)\(11\!\cdots\!54\)\( T^{10} + \)\(17\!\cdots\!32\)\( T^{11} + \)\(16\!\cdots\!01\)\( T^{12} + \)\(17\!\cdots\!32\)\( p^{3} T^{13} - \)\(11\!\cdots\!54\)\( p^{6} T^{14} - \)\(30\!\cdots\!76\)\( p^{9} T^{15} + 22258041301325289726 p^{12} T^{16} - 152529932331002772 p^{15} T^{17} - 585614628341414 p^{18} T^{18} + 4649175328776 p^{21} T^{19} + 17587396185 p^{24} T^{20} - 11259248 p^{27} T^{21} - 194538 p^{30} T^{22} - 156 p^{33} T^{23} + p^{36} T^{24} \) | |
| 47 | \( 1 + 462 T - 45963 T^{2} - 125217186 T^{3} - 37763799354 T^{4} + 3473731732278 T^{5} + 7013559465756311 T^{6} + 1919349579561105582 T^{7} - \)\(19\!\cdots\!40\)\( T^{8} - \)\(28\!\cdots\!26\)\( T^{9} - \)\(55\!\cdots\!47\)\( T^{10} + \)\(11\!\cdots\!38\)\( T^{11} + \)\(93\!\cdots\!12\)\( T^{12} + \)\(11\!\cdots\!38\)\( p^{3} T^{13} - \)\(55\!\cdots\!47\)\( p^{6} T^{14} - \)\(28\!\cdots\!26\)\( p^{9} T^{15} - \)\(19\!\cdots\!40\)\( p^{12} T^{16} + 1919349579561105582 p^{15} T^{17} + 7013559465756311 p^{18} T^{18} + 3473731732278 p^{21} T^{19} - 37763799354 p^{24} T^{20} - 125217186 p^{27} T^{21} - 45963 p^{30} T^{22} + 462 p^{33} T^{23} + p^{36} T^{24} \) | |
| 53 | \( ( 1 - 1014 T + 1090827 T^{2} - 693677562 T^{3} + 431162734311 T^{4} - 196855708959672 T^{5} + 86649460990723294 T^{6} - 196855708959672 p^{3} T^{7} + 431162734311 p^{6} T^{8} - 693677562 p^{9} T^{9} + 1090827 p^{12} T^{10} - 1014 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 59 | \( 1 + 1008 T - 202329 T^{2} - 322787952 T^{3} + 166526239419 T^{4} + 120425759506656 T^{5} - 41348228441440684 T^{6} - 21223519472950002432 T^{7} + \)\(79\!\cdots\!77\)\( T^{8} + \)\(23\!\cdots\!76\)\( T^{9} - \)\(10\!\cdots\!27\)\( T^{10} - \)\(23\!\cdots\!20\)\( T^{11} + \)\(47\!\cdots\!06\)\( T^{12} - \)\(23\!\cdots\!20\)\( p^{3} T^{13} - \)\(10\!\cdots\!27\)\( p^{6} T^{14} + \)\(23\!\cdots\!76\)\( p^{9} T^{15} + \)\(79\!\cdots\!77\)\( p^{12} T^{16} - 21223519472950002432 p^{15} T^{17} - 41348228441440684 p^{18} T^{18} + 120425759506656 p^{21} T^{19} + 166526239419 p^{24} T^{20} - 322787952 p^{27} T^{21} - 202329 p^{30} T^{22} + 1008 p^{33} T^{23} + p^{36} T^{24} \) | |
| 61 | \( 1 + 36 T - 639558 T^{2} - 297010160 T^{3} + 174794621373 T^{4} + 157515603387912 T^{5} - 1340629839700514 T^{6} - 36389935790699803884 T^{7} - \)\(86\!\cdots\!94\)\( T^{8} + \)\(38\!\cdots\!88\)\( T^{9} + \)\(17\!\cdots\!10\)\( T^{10} - \)\(73\!\cdots\!68\)\( T^{11} - \)\(31\!\cdots\!63\)\( T^{12} - \)\(73\!\cdots\!68\)\( p^{3} T^{13} + \)\(17\!\cdots\!10\)\( p^{6} T^{14} + \)\(38\!\cdots\!88\)\( p^{9} T^{15} - \)\(86\!\cdots\!94\)\( p^{12} T^{16} - 36389935790699803884 p^{15} T^{17} - 1340629839700514 p^{18} T^{18} + 157515603387912 p^{21} T^{19} + 174794621373 p^{24} T^{20} - 297010160 p^{27} T^{21} - 639558 p^{30} T^{22} + 36 p^{33} T^{23} + p^{36} T^{24} \) | |
| 67 | \( 1 + 144 T - 901674 T^{2} - 104137280 T^{3} + 358182156861 T^{4} + 32948354338848 T^{5} - 101401531068296654 T^{6} - 13365064468941257808 T^{7} + \)\(28\!\cdots\!30\)\( T^{8} + \)\(60\!\cdots\!04\)\( T^{9} - \)\(60\!\cdots\!82\)\( T^{10} - \)\(10\!\cdots\!88\)\( T^{11} + \)\(12\!\cdots\!45\)\( T^{12} - \)\(10\!\cdots\!88\)\( p^{3} T^{13} - \)\(60\!\cdots\!82\)\( p^{6} T^{14} + \)\(60\!\cdots\!04\)\( p^{9} T^{15} + \)\(28\!\cdots\!30\)\( p^{12} T^{16} - 13365064468941257808 p^{15} T^{17} - 101401531068296654 p^{18} T^{18} + 32948354338848 p^{21} T^{19} + 358182156861 p^{24} T^{20} - 104137280 p^{27} T^{21} - 901674 p^{30} T^{22} + 144 p^{33} T^{23} + p^{36} T^{24} \) | |
| 71 | \( ( 1 - 1212 T + 1394184 T^{2} - 941631468 T^{3} + 678486608952 T^{4} - 324733416162252 T^{5} + 216567275384376754 T^{6} - 324733416162252 p^{3} T^{7} + 678486608952 p^{6} T^{8} - 941631468 p^{9} T^{9} + 1394184 p^{12} T^{10} - 1212 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 73 | \( ( 1 + 900 T + 544194 T^{2} + 29232860 T^{3} + 118342881891 T^{4} + 87050307699360 T^{5} + 113916159378575988 T^{6} + 87050307699360 p^{3} T^{7} + 118342881891 p^{6} T^{8} + 29232860 p^{9} T^{9} + 544194 p^{12} T^{10} + 900 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 79 | \( 1 - 936 T - 417510 T^{2} + 697576960 T^{3} - 339323061771 T^{4} + 210755345803920 T^{5} - 17346353923612178 T^{6} - \)\(23\!\cdots\!48\)\( T^{7} + \)\(23\!\cdots\!86\)\( T^{8} - \)\(91\!\cdots\!84\)\( T^{9} + \)\(56\!\cdots\!66\)\( T^{10} + \)\(59\!\cdots\!04\)\( T^{11} - \)\(71\!\cdots\!31\)\( T^{12} + \)\(59\!\cdots\!04\)\( p^{3} T^{13} + \)\(56\!\cdots\!66\)\( p^{6} T^{14} - \)\(91\!\cdots\!84\)\( p^{9} T^{15} + \)\(23\!\cdots\!86\)\( p^{12} T^{16} - \)\(23\!\cdots\!48\)\( p^{15} T^{17} - 17346353923612178 p^{18} T^{18} + 210755345803920 p^{21} T^{19} - 339323061771 p^{24} T^{20} + 697576960 p^{27} T^{21} - 417510 p^{30} T^{22} - 936 p^{33} T^{23} + p^{36} T^{24} \) | |
| 83 | \( 1 + 288 T - 2252358 T^{2} - 1316699136 T^{3} + 2565198294345 T^{4} + 24069490629984 p T^{5} - 1775551807272281386 T^{6} - \)\(16\!\cdots\!24\)\( T^{7} + \)\(87\!\cdots\!70\)\( T^{8} + \)\(86\!\cdots\!88\)\( T^{9} - \)\(36\!\cdots\!90\)\( T^{10} - \)\(19\!\cdots\!56\)\( T^{11} + \)\(18\!\cdots\!77\)\( T^{12} - \)\(19\!\cdots\!56\)\( p^{3} T^{13} - \)\(36\!\cdots\!90\)\( p^{6} T^{14} + \)\(86\!\cdots\!88\)\( p^{9} T^{15} + \)\(87\!\cdots\!70\)\( p^{12} T^{16} - \)\(16\!\cdots\!24\)\( p^{15} T^{17} - 1775551807272281386 p^{18} T^{18} + 24069490629984 p^{22} T^{19} + 2565198294345 p^{24} T^{20} - 1316699136 p^{27} T^{21} - 2252358 p^{30} T^{22} + 288 p^{33} T^{23} + p^{36} T^{24} \) | |
| 89 | \( ( 1 + 120 T + 2598672 T^{2} + 262439400 T^{3} + 3155299319904 T^{4} + 236215309795320 T^{5} + 2566664385763147234 T^{6} + 236215309795320 p^{3} T^{7} + 3155299319904 p^{6} T^{8} + 262439400 p^{9} T^{9} + 2598672 p^{12} T^{10} + 120 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 97 | \( 1 - 1188 T - 3174126 T^{2} + 2504535808 T^{3} + 7411193865201 T^{4} - 3074795654613264 T^{5} - 11819367107267340050 T^{6} + \)\(19\!\cdots\!36\)\( T^{7} + \)\(14\!\cdots\!50\)\( T^{8} - \)\(35\!\cdots\!52\)\( T^{9} - \)\(14\!\cdots\!42\)\( T^{10} - \)\(14\!\cdots\!60\)\( T^{11} + \)\(13\!\cdots\!97\)\( T^{12} - \)\(14\!\cdots\!60\)\( p^{3} T^{13} - \)\(14\!\cdots\!42\)\( p^{6} T^{14} - \)\(35\!\cdots\!52\)\( p^{9} T^{15} + \)\(14\!\cdots\!50\)\( p^{12} T^{16} + \)\(19\!\cdots\!36\)\( p^{15} T^{17} - 11819367107267340050 p^{18} T^{18} - 3074795654613264 p^{21} T^{19} + 7411193865201 p^{24} T^{20} + 2504535808 p^{27} T^{21} - 3174126 p^{30} T^{22} - 1188 p^{33} T^{23} + p^{36} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.45005318647079245105553769640, −2.34955444915646132416135688164, −2.32889328724062598884439289137, −2.17621111643563351853602878442, −2.16181813028282756094442570434, −2.11488495609284122998472432847, −1.85807115313432405163774485937, −1.84449423679483391453778538992, −1.82809311099149549059113904197, −1.66795341557183549504389111161, −1.65172563317560015791939993301, −1.56735910509166126831002550183, −1.54606654370334288304071191434, −1.21347399405289743638546424089, −1.13280122362473677213353286679, −1.05279130122349785143842015386, −0.952128080944348530282458022854, −0.883425236583827624514357334208, −0.836164052138600444876357345632, −0.65206543728855847308751472942, −0.57259007859120018531327306795, −0.50278370394238192949693730733, −0.12315630611638226828627507252, −0.11169749640520186821544376728, −0.07362713336006339033748468749, 0.07362713336006339033748468749, 0.11169749640520186821544376728, 0.12315630611638226828627507252, 0.50278370394238192949693730733, 0.57259007859120018531327306795, 0.65206543728855847308751472942, 0.836164052138600444876357345632, 0.883425236583827624514357334208, 0.952128080944348530282458022854, 1.05279130122349785143842015386, 1.13280122362473677213353286679, 1.21347399405289743638546424089, 1.54606654370334288304071191434, 1.56735910509166126831002550183, 1.65172563317560015791939993301, 1.66795341557183549504389111161, 1.82809311099149549059113904197, 1.84449423679483391453778538992, 1.85807115313432405163774485937, 2.11488495609284122998472432847, 2.16181813028282756094442570434, 2.17621111643563351853602878442, 2.32889328724062598884439289137, 2.34955444915646132416135688164, 2.45005318647079245105553769640
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.