Dirichlet series
| L(s) = 1 | − 2·4-s + 6·5-s − 12·7-s − 3·8-s + 4·11-s − 2·13-s + 6·16-s + 17-s + 7·19-s − 12·20-s − 19·23-s + 20·25-s + 24·28-s + 29-s + 13·31-s + 10·32-s − 72·35-s + 8·37-s − 18·40-s − 8·41-s − 4·43-s − 8·44-s + 13·47-s + 16·49-s + 4·52-s − 44·53-s + 24·55-s + ⋯ |
| L(s) = 1 | − 4-s + 2.68·5-s − 4.53·7-s − 1.06·8-s + 1.20·11-s − 0.554·13-s + 3/2·16-s + 0.242·17-s + 1.60·19-s − 2.68·20-s − 3.96·23-s + 4·25-s + 4.53·28-s + 0.185·29-s + 2.33·31-s + 1.76·32-s − 12.1·35-s + 1.31·37-s − 2.84·40-s − 1.24·41-s − 0.609·43-s − 1.20·44-s + 1.89·47-s + 16/7·49-s + 0.554·52-s − 6.04·53-s + 3.23·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{24} \cdot 5^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1131.13\) |
| Root analytic conductor: | \(1.34038\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{24} \cdot 5^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.5134947868\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5134947868\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 6 T + 16 T^{2} - 21 T^{3} - 4 T^{4} + 4 p^{2} T^{5} - 57 p T^{6} + 4 p^{3} T^{7} - 4 p^{2} T^{8} - 21 p^{3} T^{9} + 16 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) | |
| 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
−4.30705291871504541749918555371, −4.20142101103515936543659870176, −3.91610362688519829335797983764, −3.85356262871722661522661470272, −3.74508412511464790025259479610, −3.67998841499595474887774941723, −3.45387970012373376762701208681, −3.33988455923528886915769536355, −3.30386167669944678712746995166, −3.15668192412204880425547358634, −3.01541187740288564806278834308, −2.98262871354911658529801793235, −2.88190064410123069906962777442, −2.82687652019421388834920781073, −2.51829339386102864612133100094, −2.51046993396445143284950057804, −2.25225657451800932228711755041, −2.20500720917927950878792245225, −1.67378115233848362894739534144, −1.57710672046606996181604229614, −1.56140787561405278751136635303, −1.35034081090724117134563503197, −1.30823927321469147368419195177, −0.44626389904788896235768033506, −0.29680581732251432449195411153, 0.29680581732251432449195411153, 0.44626389904788896235768033506, 1.30823927321469147368419195177, 1.35034081090724117134563503197, 1.56140787561405278751136635303, 1.57710672046606996181604229614, 1.67378115233848362894739534144, 2.20500720917927950878792245225, 2.25225657451800932228711755041, 2.51046993396445143284950057804, 2.51829339386102864612133100094, 2.82687652019421388834920781073, 2.88190064410123069906962777442, 2.98262871354911658529801793235, 3.01541187740288564806278834308, 3.15668192412204880425547358634, 3.30386167669944678712746995166, 3.33988455923528886915769536355, 3.45387970012373376762701208681, 3.67998841499595474887774941723, 3.74508412511464790025259479610, 3.85356262871722661522661470272, 3.91610362688519829335797983764, 4.20142101103515936543659870176, 4.30705291871504541749918555371
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.