Dirichlet series
| L(s) = 1 | − 3·3-s − 2·4-s + 12·7-s − 3·8-s + 3·9-s − 4·11-s + 6·12-s + 2·13-s + 6·16-s + 17-s + 7·19-s − 36·21-s − 19·23-s + 9·24-s − 27-s − 24·28-s − 29-s + 13·31-s + 10·32-s + 12·33-s − 6·36-s − 8·37-s − 6·39-s + 8·41-s + 4·43-s + 8·44-s + 13·47-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 4-s + 4.53·7-s − 1.06·8-s + 9-s − 1.20·11-s + 1.73·12-s + 0.554·13-s + 3/2·16-s + 0.242·17-s + 1.60·19-s − 7.85·21-s − 3.96·23-s + 1.83·24-s − 0.192·27-s − 4.53·28-s − 0.185·29-s + 2.33·31-s + 1.76·32-s + 2.08·33-s − 36-s − 1.31·37-s − 0.960·39-s + 1.24·41-s + 0.609·43-s + 1.20·44-s + 1.89·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{12} \cdot 5^{36}\) |
| Sign: | $1$ |
| Analytic conductor: | \(519637.\) |
| Root analytic conductor: | \(1.73043\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{12} \cdot 5^{36} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.212028289\) |
| \(L(\frac12)\) | \(\approx\) | \(1.212028289\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{3} \) |
| 5 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} + 3 T^{3} - p T^{4} + p T^{5} - 5 T^{6} - 5 p T^{7} - 9 p T^{8} - T^{9} - p T^{10} - 11 p T^{11} + 121 T^{12} - 11 p^{2} T^{13} - p^{3} T^{14} - p^{3} T^{15} - 9 p^{5} T^{16} - 5 p^{6} T^{17} - 5 p^{6} T^{18} + p^{8} T^{19} - p^{9} T^{20} + 3 p^{9} T^{21} + p^{11} T^{22} + p^{12} T^{24} \) |
| 7 | \( ( 1 - 6 T + 46 T^{2} - 185 T^{3} + 822 T^{4} - 2435 T^{5} + 7706 T^{6} - 2435 p T^{7} + 822 p^{2} T^{8} - 185 p^{3} T^{9} + 46 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 11 | \( 1 + 4 T - 2 p T^{2} - 120 T^{3} + 58 T^{4} + 900 T^{5} + 1437 T^{6} + 406 p T^{7} - 10704 T^{8} - 67762 T^{9} + 285698 T^{10} + 117068 T^{11} - 5557163 T^{12} + 117068 p T^{13} + 285698 p^{2} T^{14} - 67762 p^{3} T^{15} - 10704 p^{4} T^{16} + 406 p^{6} T^{17} + 1437 p^{6} T^{18} + 900 p^{7} T^{19} + 58 p^{8} T^{20} - 120 p^{9} T^{21} - 2 p^{11} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 2 T - 20 T^{2} - 8 T^{3} + 236 T^{4} + 1188 T^{5} - 3563 T^{6} - 9192 T^{7} - 20810 T^{8} + 140028 T^{9} + 822700 T^{10} - 143240 p T^{11} - 6309151 T^{12} - 143240 p^{2} T^{13} + 822700 p^{2} T^{14} + 140028 p^{3} T^{15} - 20810 p^{4} T^{16} - 9192 p^{5} T^{17} - 3563 p^{6} T^{18} + 1188 p^{7} T^{19} + 236 p^{8} T^{20} - 8 p^{9} T^{21} - 20 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - T - 22 T^{2} - 27 T^{3} - 141 T^{4} + 1839 T^{5} + 11791 T^{6} + 5192 T^{7} - 114598 T^{8} - 1351839 T^{9} - 1459527 T^{10} + 15628812 T^{11} + 54801289 T^{12} + 15628812 p T^{13} - 1459527 p^{2} T^{14} - 1351839 p^{3} T^{15} - 114598 p^{4} T^{16} + 5192 p^{5} T^{17} + 11791 p^{6} T^{18} + 1839 p^{7} T^{19} - 141 p^{8} T^{20} - 27 p^{9} T^{21} - 22 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 7 T + 16 T^{2} - 122 T^{3} + 1071 T^{4} - 728 T^{5} - 5797 T^{6} - 36918 T^{7} + 65678 T^{8} + 636002 T^{9} + 4923209 T^{10} - 1363125 p T^{11} + 2333961 p T^{12} - 1363125 p^{2} T^{13} + 4923209 p^{2} T^{14} + 636002 p^{3} T^{15} + 65678 p^{4} T^{16} - 36918 p^{5} T^{17} - 5797 p^{6} T^{18} - 728 p^{7} T^{19} + 1071 p^{8} T^{20} - 122 p^{9} T^{21} + 16 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 19 T + 87 T^{2} - 664 T^{3} - 8155 T^{4} - 10802 T^{5} + 276289 T^{6} + 1712486 T^{7} - 1232028 T^{8} - 57899808 T^{9} - 214047367 T^{10} + 649484443 T^{11} + 7850412157 T^{12} + 649484443 p T^{13} - 214047367 p^{2} T^{14} - 57899808 p^{3} T^{15} - 1232028 p^{4} T^{16} + 1712486 p^{5} T^{17} + 276289 p^{6} T^{18} - 10802 p^{7} T^{19} - 8155 p^{8} T^{20} - 664 p^{9} T^{21} + 87 p^{10} T^{22} + 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + T - 75 T^{2} - 3 T^{3} + 2691 T^{4} + 6800 T^{5} - 35703 T^{6} - 577442 T^{7} - 700036 T^{8} + 20501600 T^{9} + 54541649 T^{10} - 290598314 T^{11} - 2133422469 T^{12} - 290598314 p T^{13} + 54541649 p^{2} T^{14} + 20501600 p^{3} T^{15} - 700036 p^{4} T^{16} - 577442 p^{5} T^{17} - 35703 p^{6} T^{18} + 6800 p^{7} T^{19} + 2691 p^{8} T^{20} - 3 p^{9} T^{21} - 75 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 13 T - 49 T^{2} + 1410 T^{3} - 4283 T^{4} - 42518 T^{5} + 442503 T^{6} - 1354402 T^{7} - 10366302 T^{8} + 126161252 T^{9} - 343254057 T^{10} - 2277200953 T^{11} + 23831716157 T^{12} - 2277200953 p T^{13} - 343254057 p^{2} T^{14} + 126161252 p^{3} T^{15} - 10366302 p^{4} T^{16} - 1354402 p^{5} T^{17} + 442503 p^{6} T^{18} - 42518 p^{7} T^{19} - 4283 p^{8} T^{20} + 1410 p^{9} T^{21} - 49 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 8 T - 67 T^{2} - 31 p T^{3} - 3085 T^{4} + 39031 T^{5} + 455721 T^{6} + 1535453 T^{7} - 11486553 T^{8} - 153908236 T^{9} - 464906763 T^{10} + 3103077939 T^{11} + 36758415492 T^{12} + 3103077939 p T^{13} - 464906763 p^{2} T^{14} - 153908236 p^{3} T^{15} - 11486553 p^{4} T^{16} + 1535453 p^{5} T^{17} + 455721 p^{6} T^{18} + 39031 p^{7} T^{19} - 3085 p^{8} T^{20} - 31 p^{10} T^{21} - 67 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 8 T + 56 T^{2} - 840 T^{3} + 5172 T^{4} - 38968 T^{5} + 387993 T^{6} - 1909992 T^{7} + 12984488 T^{8} - 115425908 T^{9} + 544467188 T^{10} - 3312068648 T^{11} + 28013676237 T^{12} - 3312068648 p T^{13} + 544467188 p^{2} T^{14} - 115425908 p^{3} T^{15} + 12984488 p^{4} T^{16} - 1909992 p^{5} T^{17} + 387993 p^{6} T^{18} - 38968 p^{7} T^{19} + 5172 p^{8} T^{20} - 840 p^{9} T^{21} + 56 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( ( 1 - 2 T + 167 T^{2} - 256 T^{3} + 13452 T^{4} - 15956 T^{5} + 692036 T^{6} - 15956 p T^{7} + 13452 p^{2} T^{8} - 256 p^{3} T^{9} + 167 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 - 13 T + 37 T^{2} - 325 T^{3} + 7867 T^{4} - 56088 T^{5} + 222715 T^{6} - 1098334 T^{7} + 12592498 T^{8} - 108988468 T^{9} + 281826965 T^{10} - 1078581014 T^{11} + 21055334183 T^{12} - 1078581014 p T^{13} + 281826965 p^{2} T^{14} - 108988468 p^{3} T^{15} + 12592498 p^{4} T^{16} - 1098334 p^{5} T^{17} + 222715 p^{6} T^{18} - 56088 p^{7} T^{19} + 7867 p^{8} T^{20} - 325 p^{9} T^{21} + 37 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 44 T + 817 T^{2} + 7511 T^{3} + 18675 T^{4} - 371047 T^{5} - 5505981 T^{6} - 45196759 T^{7} - 286840983 T^{8} - 1119924348 T^{9} + 5970507613 T^{10} + 163921234483 T^{11} + 1596195784652 T^{12} + 163921234483 p T^{13} + 5970507613 p^{2} T^{14} - 1119924348 p^{3} T^{15} - 286840983 p^{4} T^{16} - 45196759 p^{5} T^{17} - 5505981 p^{6} T^{18} - 371047 p^{7} T^{19} + 18675 p^{8} T^{20} + 7511 p^{9} T^{21} + 817 p^{10} T^{22} + 44 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 22 T + 36 T^{2} - 2583 T^{3} - 16419 T^{4} + 137618 T^{5} + 1667358 T^{6} - 2698622 T^{7} - 107608927 T^{8} - 155893882 T^{9} + 5010588449 T^{10} + 10213783795 T^{11} - 198548784261 T^{12} + 10213783795 p T^{13} + 5010588449 p^{2} T^{14} - 155893882 p^{3} T^{15} - 107608927 p^{4} T^{16} - 2698622 p^{5} T^{17} + 1667358 p^{6} T^{18} + 137618 p^{7} T^{19} - 16419 p^{8} T^{20} - 2583 p^{9} T^{21} + 36 p^{10} T^{22} + 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 8 T - 130 T^{2} - 1358 T^{3} + 564 T^{4} + 21392 T^{5} + 487611 T^{6} + 8007850 T^{7} - 7109096 T^{8} - 542757032 T^{9} - 551554860 T^{10} + 10321250720 T^{11} + 3604709585 T^{12} + 10321250720 p T^{13} - 551554860 p^{2} T^{14} - 542757032 p^{3} T^{15} - 7109096 p^{4} T^{16} + 8007850 p^{5} T^{17} + 487611 p^{6} T^{18} + 21392 p^{7} T^{19} + 564 p^{8} T^{20} - 1358 p^{9} T^{21} - 130 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 6 T - 172 T^{2} + 858 T^{3} + 11749 T^{4} - 57666 T^{5} - 232014 T^{6} + 4755072 T^{7} - 22912618 T^{8} - 467752054 T^{9} + 2928188158 T^{10} + 17377714672 T^{11} - 223143036976 T^{12} + 17377714672 p T^{13} + 2928188158 p^{2} T^{14} - 467752054 p^{3} T^{15} - 22912618 p^{4} T^{16} + 4755072 p^{5} T^{17} - 232014 p^{6} T^{18} - 57666 p^{7} T^{19} + 11749 p^{8} T^{20} + 858 p^{9} T^{21} - 172 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 21 T + 46 T^{2} - 1289 T^{3} + 2891 T^{4} + 3251 p T^{5} + 1592921 T^{6} - 6954484 T^{7} - 90555524 T^{8} + 1177173511 T^{9} + 16628404231 T^{10} + 5732671986 T^{11} - 764574531549 T^{12} + 5732671986 p T^{13} + 16628404231 p^{2} T^{14} + 1177173511 p^{3} T^{15} - 90555524 p^{4} T^{16} - 6954484 p^{5} T^{17} + 1592921 p^{6} T^{18} + 3251 p^{8} T^{19} + 2891 p^{8} T^{20} - 1289 p^{9} T^{21} + 46 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 16 T - 28 T^{2} + 506 T^{3} + 12385 T^{4} + 2328 T^{5} - 561526 T^{6} - 14932744 T^{7} + 69930462 T^{8} + 421338712 T^{9} + 9017966158 T^{10} - 43966609042 T^{11} - 621466327768 T^{12} - 43966609042 p T^{13} + 9017966158 p^{2} T^{14} + 421338712 p^{3} T^{15} + 69930462 p^{4} T^{16} - 14932744 p^{5} T^{17} - 561526 p^{6} T^{18} + 2328 p^{7} T^{19} + 12385 p^{8} T^{20} + 506 p^{9} T^{21} - 28 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 10 T - 272 T^{2} + 2840 T^{3} + 26866 T^{4} - 289420 T^{5} - 489675 T^{6} + 1858730 T^{7} - 91007200 T^{8} + 1966674780 T^{9} + 3705819298 T^{10} - 105092388950 T^{11} + 146277106069 T^{12} - 105092388950 p T^{13} + 3705819298 p^{2} T^{14} + 1966674780 p^{3} T^{15} - 91007200 p^{4} T^{16} + 1858730 p^{5} T^{17} - 489675 p^{6} T^{18} - 289420 p^{7} T^{19} + 26866 p^{8} T^{20} + 2840 p^{9} T^{21} - 272 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 10 T - 32 T^{2} + 586 T^{3} + 7028 T^{4} - 84104 T^{5} - 240235 T^{6} + 5165580 T^{7} + 16282512 T^{8} - 55940648 T^{9} - 5964955348 T^{10} + 18358343086 T^{11} + 292581600901 T^{12} + 18358343086 p T^{13} - 5964955348 p^{2} T^{14} - 55940648 p^{3} T^{15} + 16282512 p^{4} T^{16} + 5165580 p^{5} T^{17} - 240235 p^{6} T^{18} - 84104 p^{7} T^{19} + 7028 p^{8} T^{20} + 586 p^{9} T^{21} - 32 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 57 T + 1416 T^{2} - 19412 T^{3} + 146681 T^{4} - 436608 T^{5} + 1888 T^{6} - 50698623 T^{7} + 1130538888 T^{8} - 10860338043 T^{9} + 55652919649 T^{10} - 136415194025 T^{11} + 148330818154 T^{12} - 136415194025 p T^{13} + 55652919649 p^{2} T^{14} - 10860338043 p^{3} T^{15} + 1130538888 p^{4} T^{16} - 50698623 p^{5} T^{17} + 1888 p^{6} T^{18} - 436608 p^{7} T^{19} + 146681 p^{8} T^{20} - 19412 p^{9} T^{21} + 1416 p^{10} T^{22} - 57 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 4 T - 152 T^{2} + 258 T^{3} + 15244 T^{4} - 16576 T^{5} - 519489 T^{6} + 1347552 T^{7} - 32374768 T^{8} - 207718764 T^{9} + 20086873388 T^{10} + 47224426932 T^{11} - 2473927275691 T^{12} + 47224426932 p T^{13} + 20086873388 p^{2} T^{14} - 207718764 p^{3} T^{15} - 32374768 p^{4} T^{16} + 1347552 p^{5} T^{17} - 519489 p^{6} T^{18} - 16576 p^{7} T^{19} + 15244 p^{8} T^{20} + 258 p^{9} T^{21} - 152 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−4.03038692839700994861919725122, −3.59986952673953312358041495062, −3.59208319667484890899255887414, −3.41886538652666908726028435938, −3.38215073980047135734573719142, −3.31545567831214501352623679169, −3.31269693182740360229046748109, −3.14005959187795652608698013158, −3.06413544775244734044303775198, −2.87945960614785125193004372888, −2.69339067349197327009847472792, −2.47056822529885328366454577796, −2.39006180444384902046441239365, −2.27538233188217695799508256650, −2.16144580836999926273144312560, −1.81422051195478972686516118983, −1.68647252139506365619481169771, −1.67087400574516844321469803076, −1.60883599397637330483279363547, −1.55042613542368591587310148864, −1.27036380497927351049979602389, −1.21289516569213679204606912059, −0.71642549981840436284931786188, −0.48066038901943911894846694573, −0.27562050985500005229510225076, 0.27562050985500005229510225076, 0.48066038901943911894846694573, 0.71642549981840436284931786188, 1.21289516569213679204606912059, 1.27036380497927351049979602389, 1.55042613542368591587310148864, 1.60883599397637330483279363547, 1.67087400574516844321469803076, 1.68647252139506365619481169771, 1.81422051195478972686516118983, 2.16144580836999926273144312560, 2.27538233188217695799508256650, 2.39006180444384902046441239365, 2.47056822529885328366454577796, 2.69339067349197327009847472792, 2.87945960614785125193004372888, 3.06413544775244734044303775198, 3.14005959187795652608698013158, 3.31269693182740360229046748109, 3.31545567831214501352623679169, 3.38215073980047135734573719142, 3.41886538652666908726028435938, 3.59208319667484890899255887414, 3.59986952673953312358041495062, 4.03038692839700994861919725122
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.