Dirichlet series
| L(s) = 1 | − 3·2-s + 6·4-s + 3·5-s − 6·7-s − 9·8-s − 9·10-s + 6·11-s − 6·13-s + 18·14-s + 12·16-s − 18·17-s + 24·19-s + 18·20-s − 18·22-s + 12·23-s + 15·25-s + 18·26-s − 36·28-s − 21·29-s − 15·31-s − 12·32-s + 54·34-s − 18·35-s + 6·37-s − 72·38-s − 27·40-s + 12·41-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s + 1.34·5-s − 2.26·7-s − 3.18·8-s − 2.84·10-s + 1.80·11-s − 1.66·13-s + 4.81·14-s + 3·16-s − 4.36·17-s + 5.50·19-s + 4.02·20-s − 3.83·22-s + 2.50·23-s + 3·25-s + 3.53·26-s − 6.80·28-s − 3.89·29-s − 2.69·31-s − 2.12·32-s + 9.26·34-s − 3.04·35-s + 0.986·37-s − 11.6·38-s − 4.26·40-s + 1.87·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{72}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.51375\times 10^{9}\) |
| Root analytic conductor: | \(2.41269\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{72} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.423823141\) |
| \(L(\frac12)\) | \(\approx\) | \(1.423823141\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 3 T + 3 T^{2} - 3 T^{4} - 3 p T^{5} - T^{6} + 27 T^{7} + 27 p T^{8} + 27 p T^{9} + 9 p^{2} T^{10} - 9 p^{2} T^{11} - 135 T^{12} - 9 p^{3} T^{13} + 9 p^{4} T^{14} + 27 p^{4} T^{15} + 27 p^{5} T^{16} + 27 p^{5} T^{17} - p^{6} T^{18} - 3 p^{8} T^{19} - 3 p^{8} T^{20} + 3 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 - 3 T - 6 T^{2} - 9 T^{3} + 87 T^{4} + 6 p^{2} T^{5} - 38 p T^{6} - 909 T^{7} - 1521 T^{8} + 2808 T^{9} + 891 p T^{10} + 3969 T^{11} - 19179 T^{12} + 3969 p T^{13} + 891 p^{3} T^{14} + 2808 p^{3} T^{15} - 1521 p^{4} T^{16} - 909 p^{5} T^{17} - 38 p^{7} T^{18} + 6 p^{9} T^{19} + 87 p^{8} T^{20} - 9 p^{9} T^{21} - 6 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 + 6 T + 3 T^{2} - 8 p T^{3} - 150 T^{4} - 48 T^{5} + 10 T^{6} - 1440 T^{7} - 477 T^{8} + 20932 T^{9} + 52197 T^{10} - 25890 T^{11} - 306689 T^{12} - 25890 p T^{13} + 52197 p^{2} T^{14} + 20932 p^{3} T^{15} - 477 p^{4} T^{16} - 1440 p^{5} T^{17} + 10 p^{6} T^{18} - 48 p^{7} T^{19} - 150 p^{8} T^{20} - 8 p^{10} T^{21} + 3 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 6 T - 15 T^{2} + 72 T^{3} + 402 T^{4} - 114 T^{5} - 8848 T^{6} + 7092 T^{7} + 73287 T^{8} - 29538 T^{9} - 873405 T^{10} - 573048 T^{11} + 15237891 T^{12} - 573048 p T^{13} - 873405 p^{2} T^{14} - 29538 p^{3} T^{15} + 73287 p^{4} T^{16} + 7092 p^{5} T^{17} - 8848 p^{6} T^{18} - 114 p^{7} T^{19} + 402 p^{8} T^{20} + 72 p^{9} T^{21} - 15 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 6 T - 24 T^{2} - 272 T^{3} + 30 T^{4} + 6234 T^{5} + 10342 T^{6} - 93456 T^{7} - 311490 T^{8} + 964312 T^{9} + 5908470 T^{10} - 4890696 T^{11} - 86990081 T^{12} - 4890696 p T^{13} + 5908470 p^{2} T^{14} + 964312 p^{3} T^{15} - 311490 p^{4} T^{16} - 93456 p^{5} T^{17} + 10342 p^{6} T^{18} + 6234 p^{7} T^{19} + 30 p^{8} T^{20} - 272 p^{9} T^{21} - 24 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( ( 1 + 9 T + 6 p T^{2} + 630 T^{3} + 4227 T^{4} + 19611 T^{5} + 5591 p T^{6} + 19611 p T^{7} + 4227 p^{2} T^{8} + 630 p^{3} T^{9} + 6 p^{5} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( ( 1 - 12 T + 129 T^{2} - 985 T^{3} + 6471 T^{4} - 35073 T^{5} + 166182 T^{6} - 35073 p T^{7} + 6471 p^{2} T^{8} - 985 p^{3} T^{9} + 129 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 23 | \( 1 - 12 T - 15 T^{2} + 450 T^{3} + 2022 T^{4} - 18012 T^{5} - 97840 T^{6} + 468468 T^{7} + 3500631 T^{8} - 7484670 T^{9} - 108075537 T^{10} + 86356332 T^{11} + 2494857051 T^{12} + 86356332 p T^{13} - 108075537 p^{2} T^{14} - 7484670 p^{3} T^{15} + 3500631 p^{4} T^{16} + 468468 p^{5} T^{17} - 97840 p^{6} T^{18} - 18012 p^{7} T^{19} + 2022 p^{8} T^{20} + 450 p^{9} T^{21} - 15 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 21 T + 156 T^{2} + 333 T^{3} - 1335 T^{4} - 11436 T^{5} - 99298 T^{6} - 512379 T^{7} + 1250271 T^{8} + 18237096 T^{9} + 38476611 T^{10} + 247254327 T^{11} + 2704348845 T^{12} + 247254327 p T^{13} + 38476611 p^{2} T^{14} + 18237096 p^{3} T^{15} + 1250271 p^{4} T^{16} - 512379 p^{5} T^{17} - 99298 p^{6} T^{18} - 11436 p^{7} T^{19} - 1335 p^{8} T^{20} + 333 p^{9} T^{21} + 156 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 15 T - 6 T^{2} - 731 T^{3} + 3234 T^{4} + 45879 T^{5} - 179792 T^{6} - 1417725 T^{7} + 10709046 T^{8} + 30200209 T^{9} - 495230610 T^{10} - 533330277 T^{11} + 15229407838 T^{12} - 533330277 p T^{13} - 495230610 p^{2} T^{14} + 30200209 p^{3} T^{15} + 10709046 p^{4} T^{16} - 1417725 p^{5} T^{17} - 179792 p^{6} T^{18} + 45879 p^{7} T^{19} + 3234 p^{8} T^{20} - 731 p^{9} T^{21} - 6 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( ( 1 - 3 T + 93 T^{2} - 94 T^{3} + 4608 T^{4} + 864 T^{5} + 171555 T^{6} + 864 p T^{7} + 4608 p^{2} T^{8} - 94 p^{3} T^{9} + 93 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 - 12 T - 42 T^{2} + 990 T^{3} + 15 T^{4} - 44292 T^{5} + 3617 T^{6} + 2617299 T^{7} - 8664444 T^{8} - 83406402 T^{9} + 621970299 T^{10} + 647096049 T^{11} - 21957305292 T^{12} + 647096049 p T^{13} + 621970299 p^{2} T^{14} - 83406402 p^{3} T^{15} - 8664444 p^{4} T^{16} + 2617299 p^{5} T^{17} + 3617 p^{6} T^{18} - 44292 p^{7} T^{19} + 15 p^{8} T^{20} + 990 p^{9} T^{21} - 42 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 6 T - 114 T^{2} - 416 T^{3} + 7617 T^{4} + 10428 T^{5} - 340838 T^{6} - 469998 T^{7} + 8946594 T^{8} + 23651170 T^{9} - 277922874 T^{10} - 369502188 T^{11} + 14787611473 T^{12} - 369502188 p T^{13} - 277922874 p^{2} T^{14} + 23651170 p^{3} T^{15} + 8946594 p^{4} T^{16} - 469998 p^{5} T^{17} - 340838 p^{6} T^{18} + 10428 p^{7} T^{19} + 7617 p^{8} T^{20} - 416 p^{9} T^{21} - 114 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 15 T - 15 T^{2} + 1332 T^{3} - 4053 T^{4} - 50577 T^{5} + 149399 T^{6} + 2935773 T^{7} - 11979270 T^{8} - 128353842 T^{9} + 985854933 T^{10} + 1739075679 T^{11} - 45888436269 T^{12} + 1739075679 p T^{13} + 985854933 p^{2} T^{14} - 128353842 p^{3} T^{15} - 11979270 p^{4} T^{16} + 2935773 p^{5} T^{17} + 149399 p^{6} T^{18} - 50577 p^{7} T^{19} - 4053 p^{8} T^{20} + 1332 p^{9} T^{21} - 15 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 9 T + 237 T^{2} + 1656 T^{3} + 26583 T^{4} + 151479 T^{5} + 1786030 T^{6} + 151479 p T^{7} + 26583 p^{2} T^{8} + 1656 p^{3} T^{9} + 237 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 + 6 T - 213 T^{2} - 1296 T^{3} + 25512 T^{4} + 153816 T^{5} - 1929232 T^{6} - 11972142 T^{7} + 97225533 T^{8} + 608531454 T^{9} - 3222929547 T^{10} - 14424181056 T^{11} + 112233818043 T^{12} - 14424181056 p T^{13} - 3222929547 p^{2} T^{14} + 608531454 p^{3} T^{15} + 97225533 p^{4} T^{16} - 11972142 p^{5} T^{17} - 1929232 p^{6} T^{18} + 153816 p^{7} T^{19} + 25512 p^{8} T^{20} - 1296 p^{9} T^{21} - 213 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 24 T + 93 T^{2} - 1946 T^{3} - 10266 T^{4} + 172446 T^{5} + 1060300 T^{6} - 9090504 T^{7} - 44005779 T^{8} + 700679680 T^{9} + 78417087 p T^{10} - 13840719090 T^{11} - 283034464943 T^{12} - 13840719090 p T^{13} + 78417087 p^{3} T^{14} + 700679680 p^{3} T^{15} - 44005779 p^{4} T^{16} - 9090504 p^{5} T^{17} + 1060300 p^{6} T^{18} + 172446 p^{7} T^{19} - 10266 p^{8} T^{20} - 1946 p^{9} T^{21} + 93 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 15 T - 123 T^{2} - 2792 T^{3} + 6357 T^{4} + 236553 T^{5} - 637361 T^{6} - 187659 p T^{7} + 114802740 T^{8} + 538384030 T^{9} - 12717543819 T^{10} - 12718124403 T^{11} + 988677433003 T^{12} - 12718124403 p T^{13} - 12717543819 p^{2} T^{14} + 538384030 p^{3} T^{15} + 114802740 p^{4} T^{16} - 187659 p^{6} T^{17} - 637361 p^{6} T^{18} + 236553 p^{7} T^{19} + 6357 p^{8} T^{20} - 2792 p^{9} T^{21} - 123 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( ( 1 + 246 T^{2} - 864 T^{3} + 27087 T^{4} - 163296 T^{5} + 2111380 T^{6} - 163296 p T^{7} + 27087 p^{2} T^{8} - 864 p^{3} T^{9} + 246 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 73 | \( ( 1 - 12 T + 282 T^{2} - 2326 T^{3} + 34542 T^{4} - 226674 T^{5} + 2873211 T^{6} - 226674 p T^{7} + 34542 p^{2} T^{8} - 2326 p^{3} T^{9} + 282 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 79 | \( 1 + 24 T + 75 T^{2} - 1586 T^{3} + 9156 T^{4} + 425598 T^{5} + 2207908 T^{6} - 21534534 T^{7} - 220820679 T^{8} + 1920297382 T^{9} + 32423493453 T^{10} - 57642789732 T^{11} - 2974107069509 T^{12} - 57642789732 p T^{13} + 32423493453 p^{2} T^{14} + 1920297382 p^{3} T^{15} - 220820679 p^{4} T^{16} - 21534534 p^{5} T^{17} + 2207908 p^{6} T^{18} + 425598 p^{7} T^{19} + 9156 p^{8} T^{20} - 1586 p^{9} T^{21} + 75 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 6 T - 375 T^{2} + 1512 T^{3} + 81582 T^{4} - 207186 T^{5} - 12797956 T^{6} + 19690128 T^{7} + 1584961659 T^{8} - 1269102870 T^{9} - 164608491789 T^{10} + 478472652 p T^{11} + 14685582800691 T^{12} + 478472652 p^{2} T^{13} - 164608491789 p^{2} T^{14} - 1269102870 p^{3} T^{15} + 1584961659 p^{4} T^{16} + 19690128 p^{5} T^{17} - 12797956 p^{6} T^{18} - 207186 p^{7} T^{19} + 81582 p^{8} T^{20} + 1512 p^{9} T^{21} - 375 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( ( 1 + 9 T + 309 T^{2} + 3006 T^{3} + 49110 T^{4} + 463734 T^{5} + 5132383 T^{6} + 463734 p T^{7} + 49110 p^{2} T^{8} + 3006 p^{3} T^{9} + 309 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 - 21 T - 213 T^{2} + 4318 T^{3} + 79581 T^{4} - 852789 T^{5} - 15536735 T^{6} + 93324870 T^{7} + 2428778718 T^{8} - 6590604818 T^{9} - 309431633322 T^{10} + 305897810664 T^{11} + 31210720717540 T^{12} + 305897810664 p T^{13} - 309431633322 p^{2} T^{14} - 6590604818 p^{3} T^{15} + 2428778718 p^{4} T^{16} + 93324870 p^{5} T^{17} - 15536735 p^{6} T^{18} - 852789 p^{7} T^{19} + 79581 p^{8} T^{20} + 4318 p^{9} T^{21} - 213 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.17410019866352052897629386414, −3.17195365667453867357634180174, −3.15854857074466871329998084594, −3.09039465215261436143747562579, −2.88879389064029771475394957812, −2.85493689510042416067771745417, −2.65357760676528232762852293544, −2.63038763361332788757034121486, −2.47664318401442546730100946568, −2.45236387474715049476445580038, −2.38331064328554843556012108609, −2.35949029598725030653252059206, −1.98984389683490643987638514683, −1.87330447859963838101780506244, −1.77767826218015488202376326621, −1.64027894837032147625556708545, −1.53965201446149950478751002517, −1.41892294634490554588762506530, −1.29830692185635234190230551306, −1.15042868295493676050340453188, −1.06777536318680063592018764150, −0.915458701338314574673125688764, −0.41377968883626438970417442863, −0.36808030650354557377827098369, −0.28304867761566452197403852949, 0.28304867761566452197403852949, 0.36808030650354557377827098369, 0.41377968883626438970417442863, 0.915458701338314574673125688764, 1.06777536318680063592018764150, 1.15042868295493676050340453188, 1.29830692185635234190230551306, 1.41892294634490554588762506530, 1.53965201446149950478751002517, 1.64027894837032147625556708545, 1.77767826218015488202376326621, 1.87330447859963838101780506244, 1.98984389683490643987638514683, 2.35949029598725030653252059206, 2.38331064328554843556012108609, 2.45236387474715049476445580038, 2.47664318401442546730100946568, 2.63038763361332788757034121486, 2.65357760676528232762852293544, 2.85493689510042416067771745417, 2.88879389064029771475394957812, 3.09039465215261436143747562579, 3.15854857074466871329998084594, 3.17195365667453867357634180174, 3.17410019866352052897629386414
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.