Dirichlet series
| L(s) = 1 | + 8·5-s + 16·7-s + 32·11-s + 4·13-s + 28·17-s + 80·23-s + 30·25-s − 128·31-s + 128·35-s + 92·37-s − 144·41-s − 64·43-s + 80·47-s + 128·49-s − 100·53-s + 256·55-s − 240·61-s + 32·65-s − 160·67-s − 288·71-s − 4·73-s + 512·77-s − 27·81-s − 256·83-s + 224·85-s + 64·91-s + 220·97-s + ⋯ |
| L(s) = 1 | + 8/5·5-s + 16/7·7-s + 2.90·11-s + 4/13·13-s + 1.64·17-s + 3.47·23-s + 6/5·25-s − 4.12·31-s + 3.65·35-s + 2.48·37-s − 3.51·41-s − 1.48·43-s + 1.70·47-s + 2.61·49-s − 1.88·53-s + 4.65·55-s − 3.93·61-s + 0.492·65-s − 2.38·67-s − 4.05·71-s − 0.0547·73-s + 6.64·77-s − 1/3·81-s − 3.08·83-s + 2.63·85-s + 0.703·91-s + 2.26·97-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{72} \cdot 3^{12} \cdot 5^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.02630\times 10^{17}\) |
| Root analytic conductor: | \(5.11449\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{72} \cdot 3^{12} \cdot 5^{12} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(6.026810699\) |
| \(L(\frac12)\) | \(\approx\) | \(6.026810699\) |
| \(L(2)\) | 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 | \( ( 1 + p^{2} T^{4} )^{3} \) | |
| 5 | \( 1 - 8 T + 34 T^{2} - 152 T^{3} + 151 p T^{4} - 736 p T^{5} + 548 p^{2} T^{6} - 736 p^{3} T^{7} + 151 p^{5} T^{8} - 152 p^{6} T^{9} + 34 p^{8} T^{10} - 8 p^{10} T^{11} + p^{12} T^{12} \) | |
| good | 7 | \( 1 - 16 T + 128 T^{2} - 80 p T^{3} + 3998 T^{4} - 30640 T^{5} + 19328 p T^{6} + 898160 T^{7} - 7980737 T^{8} + 30200480 T^{9} - 36151552 T^{10} + 2877755616 T^{11} - 24051601468 T^{12} + 2877755616 p^{2} T^{13} - 36151552 p^{4} T^{14} + 30200480 p^{6} T^{15} - 7980737 p^{8} T^{16} + 898160 p^{10} T^{17} + 19328 p^{13} T^{18} - 30640 p^{14} T^{19} + 3998 p^{16} T^{20} - 80 p^{19} T^{21} + 128 p^{20} T^{22} - 16 p^{22} T^{23} + p^{24} T^{24} \) |
| 11 | \( ( 1 - 16 T + 570 T^{2} - 8144 T^{3} + 145331 T^{4} - 1798752 T^{5} + 22019284 T^{6} - 1798752 p^{2} T^{7} + 145331 p^{4} T^{8} - 8144 p^{6} T^{9} + 570 p^{8} T^{10} - 16 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 13 | \( 1 - 4 T + 8 T^{2} + 788 T^{3} - 11662 T^{4} - 18124 p T^{5} + 1346216 T^{6} - 42202388 T^{7} - 834042337 T^{8} + 9317487672 T^{9} - 20397015472 T^{10} + 247979819624 T^{11} + 23809114260252 T^{12} + 247979819624 p^{2} T^{13} - 20397015472 p^{4} T^{14} + 9317487672 p^{6} T^{15} - 834042337 p^{8} T^{16} - 42202388 p^{10} T^{17} + 1346216 p^{12} T^{18} - 18124 p^{15} T^{19} - 11662 p^{16} T^{20} + 788 p^{18} T^{21} + 8 p^{20} T^{22} - 4 p^{22} T^{23} + p^{24} T^{24} \) | |
| 17 | \( 1 - 28 T + 392 T^{2} - 820 T^{3} - 202990 T^{4} + 2658332 T^{5} + 5474984 T^{6} - 1112544812 T^{7} + 34896770495 T^{8} - 332181514680 T^{9} + 1178566636112 T^{10} + 41751417016088 T^{11} - 1751317556218788 T^{12} + 41751417016088 p^{2} T^{13} + 1178566636112 p^{4} T^{14} - 332181514680 p^{6} T^{15} + 34896770495 p^{8} T^{16} - 1112544812 p^{10} T^{17} + 5474984 p^{12} T^{18} + 2658332 p^{14} T^{19} - 202990 p^{16} T^{20} - 820 p^{18} T^{21} + 392 p^{20} T^{22} - 28 p^{22} T^{23} + p^{24} T^{24} \) | |
| 19 | \( 1 - 1756 T^{2} + 1534242 T^{4} - 866185772 T^{6} + 362947268303 T^{8} - 127555956267192 T^{10} + 44293304304617308 T^{12} - 127555956267192 p^{4} T^{14} + 362947268303 p^{8} T^{16} - 866185772 p^{12} T^{18} + 1534242 p^{16} T^{20} - 1756 p^{20} T^{22} + p^{24} T^{24} \) | |
| 23 | \( 1 - 80 T + 3200 T^{2} - 81424 T^{3} + 1719238 T^{4} - 52433488 T^{5} + 2008051328 T^{6} - 62231998864 T^{7} + 1403291540943 T^{8} - 27269266326560 T^{9} + 680503833484544 T^{10} - 21245523979771808 T^{11} + 567805420018629012 T^{12} - 21245523979771808 p^{2} T^{13} + 680503833484544 p^{4} T^{14} - 27269266326560 p^{6} T^{15} + 1403291540943 p^{8} T^{16} - 62231998864 p^{10} T^{17} + 2008051328 p^{12} T^{18} - 52433488 p^{14} T^{19} + 1719238 p^{16} T^{20} - 81424 p^{18} T^{21} + 3200 p^{20} T^{22} - 80 p^{22} T^{23} + p^{24} T^{24} \) | |
| 29 | \( 1 - 84 p T^{2} + 3501994 T^{4} - 4248462676 T^{6} + 4804204307327 T^{8} - 4757500940411048 T^{10} + 4135636370325613676 T^{12} - 4757500940411048 p^{4} T^{14} + 4804204307327 p^{8} T^{16} - 4248462676 p^{12} T^{18} + 3501994 p^{16} T^{20} - 84 p^{21} T^{22} + p^{24} T^{24} \) | |
| 31 | \( ( 1 + 64 T + 5702 T^{2} + 256064 T^{3} + 13289615 T^{4} + 450713472 T^{5} + 16787698068 T^{6} + 450713472 p^{2} T^{7} + 13289615 p^{4} T^{8} + 256064 p^{6} T^{9} + 5702 p^{8} T^{10} + 64 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 37 | \( 1 - 92 T + 4232 T^{2} - 244084 T^{3} + 13354418 T^{4} - 472532804 T^{5} + 16745620520 T^{6} - 709040738444 T^{7} + 22405788109663 T^{8} - 656784278554616 T^{9} + 19491575810000 p^{2} T^{10} - 25347130517459208 p T^{11} + 30055765593007176092 T^{12} - 25347130517459208 p^{3} T^{13} + 19491575810000 p^{6} T^{14} - 656784278554616 p^{6} T^{15} + 22405788109663 p^{8} T^{16} - 709040738444 p^{10} T^{17} + 16745620520 p^{12} T^{18} - 472532804 p^{14} T^{19} + 13354418 p^{16} T^{20} - 244084 p^{18} T^{21} + 4232 p^{20} T^{22} - 92 p^{22} T^{23} + p^{24} T^{24} \) | |
| 41 | \( ( 1 + 72 T + 5230 T^{2} + 298888 T^{3} + 15567647 T^{4} + 691727024 T^{5} + 32010713636 T^{6} + 691727024 p^{2} T^{7} + 15567647 p^{4} T^{8} + 298888 p^{6} T^{9} + 5230 p^{8} T^{10} + 72 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 43 | \( 1 + 64 T + 2048 T^{2} + 37696 T^{3} + 1208486 T^{4} + 277028416 T^{5} + 15965333504 T^{6} + 882757974848 T^{7} + 32851215314351 T^{8} + 881540017566336 T^{9} + 41494206391144448 T^{10} + 2771547045540767872 T^{11} + \)\(17\!\cdots\!24\)\( T^{12} + 2771547045540767872 p^{2} T^{13} + 41494206391144448 p^{4} T^{14} + 881540017566336 p^{6} T^{15} + 32851215314351 p^{8} T^{16} + 882757974848 p^{10} T^{17} + 15965333504 p^{12} T^{18} + 277028416 p^{14} T^{19} + 1208486 p^{16} T^{20} + 37696 p^{18} T^{21} + 2048 p^{20} T^{22} + 64 p^{22} T^{23} + p^{24} T^{24} \) | |
| 47 | \( 1 - 80 T + 3200 T^{2} + 20720 T^{3} - 361786 T^{4} - 325931600 T^{5} + 27446902400 T^{6} - 399110867600 T^{7} + 7706343875215 T^{8} - 1353938564832800 T^{9} + 138430816580230400 T^{10} - 1850938617314896800 T^{11} - 8515197787690439020 T^{12} - 1850938617314896800 p^{2} T^{13} + 138430816580230400 p^{4} T^{14} - 1353938564832800 p^{6} T^{15} + 7706343875215 p^{8} T^{16} - 399110867600 p^{10} T^{17} + 27446902400 p^{12} T^{18} - 325931600 p^{14} T^{19} - 361786 p^{16} T^{20} + 20720 p^{18} T^{21} + 3200 p^{20} T^{22} - 80 p^{22} T^{23} + p^{24} T^{24} \) | |
| 53 | \( 1 + 100 T + 5000 T^{2} - 14420 T^{3} - 15759502 T^{4} - 137171300 T^{5} + 65184348200 T^{6} + 5287167354740 T^{7} + 51520615851743 T^{8} - 10604502022829880 T^{9} - 357613386198612400 T^{10} + 29930733637158865240 T^{11} + \)\(34\!\cdots\!92\)\( T^{12} + 29930733637158865240 p^{2} T^{13} - 357613386198612400 p^{4} T^{14} - 10604502022829880 p^{6} T^{15} + 51520615851743 p^{8} T^{16} + 5287167354740 p^{10} T^{17} + 65184348200 p^{12} T^{18} - 137171300 p^{14} T^{19} - 15759502 p^{16} T^{20} - 14420 p^{18} T^{21} + 5000 p^{20} T^{22} + 100 p^{22} T^{23} + p^{24} T^{24} \) | |
| 59 | \( 1 - 22772 T^{2} + 257247114 T^{4} - 1947661488964 T^{6} + 11222974795073855 T^{8} - 52189340303518442184 T^{10} + \)\(19\!\cdots\!40\)\( T^{12} - 52189340303518442184 p^{4} T^{14} + 11222974795073855 p^{8} T^{16} - 1947661488964 p^{12} T^{18} + 257247114 p^{16} T^{20} - 22772 p^{20} T^{22} + p^{24} T^{24} \) | |
| 61 | \( ( 1 + 120 T + 19678 T^{2} + 1979768 T^{3} + 176200943 T^{4} + 13741195984 T^{5} + 869307440132 T^{6} + 13741195984 p^{2} T^{7} + 176200943 p^{4} T^{8} + 1979768 p^{6} T^{9} + 19678 p^{8} T^{10} + 120 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 67 | \( 1 + 160 T + 12800 T^{2} + 894368 T^{3} + 56302310 T^{4} + 4241886496 T^{5} + 357979331072 T^{6} + 500575772512 p T^{7} + 3350528703789679 T^{8} + 235679461215923776 T^{9} + 13659513455075947520 T^{10} + \)\(79\!\cdots\!88\)\( T^{11} + \)\(46\!\cdots\!36\)\( T^{12} + \)\(79\!\cdots\!88\)\( p^{2} T^{13} + 13659513455075947520 p^{4} T^{14} + 235679461215923776 p^{6} T^{15} + 3350528703789679 p^{8} T^{16} + 500575772512 p^{11} T^{17} + 357979331072 p^{12} T^{18} + 4241886496 p^{14} T^{19} + 56302310 p^{16} T^{20} + 894368 p^{18} T^{21} + 12800 p^{20} T^{22} + 160 p^{22} T^{23} + p^{24} T^{24} \) | |
| 71 | \( ( 1 + 144 T + 20806 T^{2} + 2147792 T^{3} + 196689455 T^{4} + 14930202016 T^{5} + 1181285698580 T^{6} + 14930202016 p^{2} T^{7} + 196689455 p^{4} T^{8} + 2147792 p^{6} T^{9} + 20806 p^{8} T^{10} + 144 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 73 | \( 1 + 4 T + 8 T^{2} + 250316 T^{3} - 27430838 T^{4} - 2301366916 T^{5} + 22343028968 T^{6} - 14453867405164 T^{7} + 171192585765519 T^{8} + 51129766501229032 T^{9} - 153851898992049136 T^{10} + 63097081741601612728 T^{11} + \)\(31\!\cdots\!56\)\( T^{12} + 63097081741601612728 p^{2} T^{13} - 153851898992049136 p^{4} T^{14} + 51129766501229032 p^{6} T^{15} + 171192585765519 p^{8} T^{16} - 14453867405164 p^{10} T^{17} + 22343028968 p^{12} T^{18} - 2301366916 p^{14} T^{19} - 27430838 p^{16} T^{20} + 250316 p^{18} T^{21} + 8 p^{20} T^{22} + 4 p^{22} T^{23} + p^{24} T^{24} \) | |
| 79 | \( 1 - 23564 T^{2} + 357486786 T^{4} - 3800372898268 T^{6} + 32750365366977647 T^{8} - \)\(23\!\cdots\!88\)\( T^{10} + \)\(15\!\cdots\!12\)\( T^{12} - \)\(23\!\cdots\!88\)\( p^{4} T^{14} + 32750365366977647 p^{8} T^{16} - 3800372898268 p^{12} T^{18} + 357486786 p^{16} T^{20} - 23564 p^{20} T^{22} + p^{24} T^{24} \) | |
| 83 | \( 1 + 256 T + 32768 T^{2} + 3965696 T^{3} + 461748454 T^{4} + 41742913280 T^{5} + 3418984841216 T^{6} + 298841828906240 T^{7} + 26331126357485295 T^{8} + 2270240834748981760 T^{9} + \)\(19\!\cdots\!76\)\( T^{10} + \)\(17\!\cdots\!20\)\( T^{11} + \)\(15\!\cdots\!60\)\( T^{12} + \)\(17\!\cdots\!20\)\( p^{2} T^{13} + \)\(19\!\cdots\!76\)\( p^{4} T^{14} + 2270240834748981760 p^{6} T^{15} + 26331126357485295 p^{8} T^{16} + 298841828906240 p^{10} T^{17} + 3418984841216 p^{12} T^{18} + 41742913280 p^{14} T^{19} + 461748454 p^{16} T^{20} + 3965696 p^{18} T^{21} + 32768 p^{20} T^{22} + 256 p^{22} T^{23} + p^{24} T^{24} \) | |
| 89 | \( 1 - 68476 T^{2} + 2293591362 T^{4} - 49488875939852 T^{6} + 764416368755168303 T^{8} - \)\(88\!\cdots\!12\)\( T^{10} + \)\(79\!\cdots\!68\)\( T^{12} - \)\(88\!\cdots\!12\)\( p^{4} T^{14} + 764416368755168303 p^{8} T^{16} - 49488875939852 p^{12} T^{18} + 2293591362 p^{16} T^{20} - 68476 p^{20} T^{22} + p^{24} T^{24} \) | |
| 97 | \( 1 - 220 T + 24200 T^{2} - 1617428 T^{3} + 88607242 T^{4} - 7737098660 T^{5} + 865903116392 T^{6} - 49647210459020 T^{7} + 1051808246059215 T^{8} + 492398717580431144 T^{9} - \)\(10\!\cdots\!84\)\( T^{10} + \)\(16\!\cdots\!64\)\( T^{11} - \)\(19\!\cdots\!32\)\( T^{12} + \)\(16\!\cdots\!64\)\( p^{2} T^{13} - \)\(10\!\cdots\!84\)\( p^{4} T^{14} + 492398717580431144 p^{6} T^{15} + 1051808246059215 p^{8} T^{16} - 49647210459020 p^{10} T^{17} + 865903116392 p^{12} T^{18} - 7737098660 p^{14} T^{19} + 88607242 p^{16} T^{20} - 1617428 p^{18} T^{21} + 24200 p^{20} T^{22} - 220 p^{22} T^{23} + p^{24} T^{24} \) | |
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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
−2.85786818413728177624429129741, −2.80234295784442803823110245081, −2.79564720525715068328374027717, −2.73859586618158889536238764429, −2.68886601503972202139445610243, −2.47952845209105947484822307854, −2.44598380398102519332982797117, −2.12691116069529725142234767946, −1.98453609921692523211360829404, −1.95900482777732987287763286396, −1.81131238376415904695580497393, −1.72735094393545772326257980906, −1.61858779223939276106665987173, −1.56496797101436878456931713230, −1.53263798876440078747452989114, −1.30689314822924983090732023240, −1.28367254938685443150829644079, −1.24528007336833373015807465821, −1.23266615666579011320842299906, −1.01062337832774666521359173347, −0.936836085416857450101986575706, −0.64970327759561318857281858186, −0.27401813965633001940133199325, −0.25346346689475379027468252259, −0.095038519782624581903360762609, 0.095038519782624581903360762609, 0.25346346689475379027468252259, 0.27401813965633001940133199325, 0.64970327759561318857281858186, 0.936836085416857450101986575706, 1.01062337832774666521359173347, 1.23266615666579011320842299906, 1.24528007336833373015807465821, 1.28367254938685443150829644079, 1.30689314822924983090732023240, 1.53263798876440078747452989114, 1.56496797101436878456931713230, 1.61858779223939276106665987173, 1.72735094393545772326257980906, 1.81131238376415904695580497393, 1.95900482777732987287763286396, 1.98453609921692523211360829404, 2.12691116069529725142234767946, 2.44598380398102519332982797117, 2.47952845209105947484822307854, 2.68886601503972202139445610243, 2.73859586618158889536238764429, 2.79564720525715068328374027717, 2.80234295784442803823110245081, 2.85786818413728177624429129741
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.