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
| \(L(1)\) | \(\approx\) | \(0.5255332515\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5255332515\) |
| \(L(\frac{3}{2})\) | 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
−3.37238969029074383001250513887, −3.23498525246339654134147386759, −3.23095991775238003884677667742, −3.09648674052432573159586412908, −3.06024297758643636099376008926, −2.93538699010000698528484654027, −2.87362636714463341065238374735, −2.79088006352780943250900595982, −2.77988233808895844721951262178, −2.66641481971031820711569172482, −2.65419853693720718136540110051, −2.31203922840097218731575869544, −2.05655809411473315253779068792, −2.00563436551980738890601090915, −1.88325628242928916777574349095, −1.79351611327971733277131532124, −1.77773585070207320184904259370, −1.52510346913649049204251592041, −1.27366781146112692728748362775, −0.967272969664452184742312483763, −0.904576978184203258710874057489, −0.861854829186206914209836506245, −0.852842285448942225754693816618, −0.55846019360582081506188357467, −0.03729868617224404585428071870, 0.03729868617224404585428071870, 0.55846019360582081506188357467, 0.852842285448942225754693816618, 0.861854829186206914209836506245, 0.904576978184203258710874057489, 0.967272969664452184742312483763, 1.27366781146112692728748362775, 1.52510346913649049204251592041, 1.77773585070207320184904259370, 1.79351611327971733277131532124, 1.88325628242928916777574349095, 2.00563436551980738890601090915, 2.05655809411473315253779068792, 2.31203922840097218731575869544, 2.65419853693720718136540110051, 2.66641481971031820711569172482, 2.77988233808895844721951262178, 2.79088006352780943250900595982, 2.87362636714463341065238374735, 2.93538699010000698528484654027, 3.06024297758643636099376008926, 3.09648674052432573159586412908, 3.23095991775238003884677667742, 3.23498525246339654134147386759, 3.37238969029074383001250513887
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.