Dirichlet series
| L(s) = 1 | − 3·2-s + 3·4-s − 6·5-s + 6·7-s + 18·10-s + 15·11-s − 3·13-s − 18·14-s − 3·16-s + 9·17-s − 12·19-s − 18·20-s − 45·22-s − 15·23-s + 12·25-s + 9·26-s + 18·28-s + 6·29-s − 12·31-s + 15·32-s − 27·34-s − 36·35-s − 3·37-s + 36·38-s + 39·41-s + 24·43-s + 45·44-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3/2·4-s − 2.68·5-s + 2.26·7-s + 5.69·10-s + 4.52·11-s − 0.832·13-s − 4.81·14-s − 3/4·16-s + 2.18·17-s − 2.75·19-s − 4.02·20-s − 9.59·22-s − 3.12·23-s + 12/5·25-s + 1.76·26-s + 3.40·28-s + 1.11·29-s − 2.15·31-s + 2.65·32-s − 4.63·34-s − 6.08·35-s − 0.493·37-s + 5.83·38-s + 6.09·41-s + 3.65·43-s + 6.78·44-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 | \( 1 + 3 T + 3 p T^{2} + 9 T^{3} + 3 p^{2} T^{4} + 3 T^{5} - 11 p T^{6} - 27 p T^{7} - 81 T^{8} - 81 T^{9} - 27 T^{10} + 117 T^{11} + 231 T^{12} + 117 p T^{13} - 27 p^{2} T^{14} - 81 p^{3} T^{15} - 81 p^{4} T^{16} - 27 p^{6} T^{17} - 11 p^{7} T^{18} + 3 p^{7} T^{19} + 3 p^{10} T^{20} + 9 p^{9} T^{21} + 3 p^{11} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 + 6 T + 24 T^{2} + 18 p T^{3} + 282 T^{4} + 834 T^{5} + 2462 T^{6} + 6831 T^{7} + 18252 T^{8} + 47034 T^{9} + 22761 p T^{10} + 264969 T^{11} + 603321 T^{12} + 264969 p T^{13} + 22761 p^{3} T^{14} + 47034 p^{3} T^{15} + 18252 p^{4} T^{16} + 6831 p^{5} T^{17} + 2462 p^{6} T^{18} + 834 p^{7} T^{19} + 282 p^{8} T^{20} + 18 p^{10} T^{21} + 24 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 - 6 T + 33 T^{2} - 125 T^{3} + 465 T^{4} - 1311 T^{5} + 79 p^{2} T^{6} - 9837 T^{7} + 26874 T^{8} - 65144 T^{9} + 177312 T^{10} - 434271 T^{11} + 1179880 T^{12} - 434271 p T^{13} + 177312 p^{2} T^{14} - 65144 p^{3} T^{15} + 26874 p^{4} T^{16} - 9837 p^{5} T^{17} + 79 p^{8} T^{18} - 1311 p^{7} T^{19} + 465 p^{8} T^{20} - 125 p^{9} T^{21} + 33 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 15 T + 96 T^{2} - 270 T^{3} - 465 T^{4} + 7761 T^{5} - 30604 T^{6} + 17064 T^{7} + 378423 T^{8} - 1824066 T^{9} + 2480751 T^{10} + 12756456 T^{11} - 76984833 T^{12} + 12756456 p T^{13} + 2480751 p^{2} T^{14} - 1824066 p^{3} T^{15} + 378423 p^{4} T^{16} + 17064 p^{5} T^{17} - 30604 p^{6} T^{18} + 7761 p^{7} T^{19} - 465 p^{8} T^{20} - 270 p^{9} T^{21} + 96 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 3 T + 51 T^{2} + 172 T^{3} + 123 p T^{4} + 5781 T^{5} + 38071 T^{6} + 136746 T^{7} + 734706 T^{8} + 2621464 T^{9} + 12062316 T^{10} + 41098578 T^{11} + 168487648 T^{12} + 41098578 p T^{13} + 12062316 p^{2} T^{14} + 2621464 p^{3} T^{15} + 734706 p^{4} T^{16} + 136746 p^{5} T^{17} + 38071 p^{6} T^{18} + 5781 p^{7} T^{19} + 123 p^{9} T^{20} + 172 p^{9} T^{21} + 51 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 9 T - 21 T^{2} + 342 T^{3} + 507 T^{4} - 7785 T^{5} - 20659 T^{6} + 148824 T^{7} + 597168 T^{8} - 1945998 T^{9} - 14374998 T^{10} + 572562 p T^{11} + 17436324 p T^{12} + 572562 p^{2} T^{13} - 14374998 p^{2} T^{14} - 1945998 p^{3} T^{15} + 597168 p^{4} T^{16} + 148824 p^{5} T^{17} - 20659 p^{6} T^{18} - 7785 p^{7} T^{19} + 507 p^{8} T^{20} + 342 p^{9} T^{21} - 21 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 12 T + 15 T^{2} - 422 T^{3} - 1650 T^{4} + 6834 T^{5} + 46954 T^{6} - 14328 T^{7} - 453285 T^{8} + 62908 T^{9} + 3171621 T^{10} + 2889474 T^{11} + 6217567 T^{12} + 2889474 p T^{13} + 3171621 p^{2} T^{14} + 62908 p^{3} T^{15} - 453285 p^{4} T^{16} - 14328 p^{5} T^{17} + 46954 p^{6} T^{18} + 6834 p^{7} T^{19} - 1650 p^{8} T^{20} - 422 p^{9} T^{21} + 15 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 15 T + 132 T^{2} + 1008 T^{3} + 6105 T^{4} + 30381 T^{5} + 142466 T^{6} + 588600 T^{7} + 2406663 T^{8} + 11882808 T^{9} + 2569275 p T^{10} + 306752526 T^{11} + 1592153511 T^{12} + 306752526 p T^{13} + 2569275 p^{3} T^{14} + 11882808 p^{3} T^{15} + 2406663 p^{4} T^{16} + 588600 p^{5} T^{17} + 142466 p^{6} T^{18} + 30381 p^{7} T^{19} + 6105 p^{8} T^{20} + 1008 p^{9} T^{21} + 132 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 6 T + 33 T^{2} - 198 T^{3} + 1074 T^{4} - 10563 T^{5} + 48587 T^{6} - 410535 T^{7} + 2604996 T^{8} - 13884210 T^{9} + 74114874 T^{10} - 348028020 T^{11} + 2336546166 T^{12} - 348028020 p T^{13} + 74114874 p^{2} T^{14} - 13884210 p^{3} T^{15} + 2604996 p^{4} T^{16} - 410535 p^{5} T^{17} + 48587 p^{6} T^{18} - 10563 p^{7} T^{19} + 1074 p^{8} T^{20} - 198 p^{9} T^{21} + 33 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 12 T + 24 T^{2} - 530 T^{3} - 6204 T^{4} - 34494 T^{5} - 17846 T^{6} + 1334250 T^{7} + 12925044 T^{8} + 61246672 T^{9} + 13581156 T^{10} - 2034793722 T^{11} - 15999983885 T^{12} - 2034793722 p T^{13} + 13581156 p^{2} T^{14} + 61246672 p^{3} T^{15} + 12925044 p^{4} T^{16} + 1334250 p^{5} T^{17} - 17846 p^{6} T^{18} - 34494 p^{7} T^{19} - 6204 p^{8} T^{20} - 530 p^{9} T^{21} + 24 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 3 T - 84 T^{2} + 91 T^{3} + 3759 T^{4} - 19770 T^{5} - 74006 T^{6} + 1076553 T^{7} - 1139895 T^{8} - 29804384 T^{9} + 197998401 T^{10} + 344437215 T^{11} - 10472509811 T^{12} + 344437215 p T^{13} + 197998401 p^{2} T^{14} - 29804384 p^{3} T^{15} - 1139895 p^{4} T^{16} + 1076553 p^{5} T^{17} - 74006 p^{6} T^{18} - 19770 p^{7} T^{19} + 3759 p^{8} T^{20} + 91 p^{9} T^{21} - 84 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 39 T + 672 T^{2} - 6390 T^{3} + 31404 T^{4} - 9741 T^{5} - 756454 T^{6} + 760050 T^{7} + 61405740 T^{8} - 623658636 T^{9} + 2184946938 T^{10} + 10024566498 T^{11} - 143612314437 T^{12} + 10024566498 p T^{13} + 2184946938 p^{2} T^{14} - 623658636 p^{3} T^{15} + 61405740 p^{4} T^{16} + 760050 p^{5} T^{17} - 756454 p^{6} T^{18} - 9741 p^{7} T^{19} + 31404 p^{8} T^{20} - 6390 p^{9} T^{21} + 672 p^{10} T^{22} - 39 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 24 T + 348 T^{2} - 3914 T^{3} + 37932 T^{4} - 340314 T^{5} + 2904454 T^{6} - 23749614 T^{7} + 183912804 T^{8} - 1372514672 T^{9} + 9999694092 T^{10} - 70491340002 T^{11} + 474808104043 T^{12} - 70491340002 p T^{13} + 9999694092 p^{2} T^{14} - 1372514672 p^{3} T^{15} + 183912804 p^{4} T^{16} - 23749614 p^{5} T^{17} + 2904454 p^{6} T^{18} - 340314 p^{7} T^{19} + 37932 p^{8} T^{20} - 3914 p^{9} T^{21} + 348 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 42 T + 798 T^{2} - 8811 T^{3} + 60429 T^{4} - 268503 T^{5} + 1243256 T^{6} - 14419053 T^{7} + 183521187 T^{8} - 1660666050 T^{9} + 10639939707 T^{10} - 51421124475 T^{11} + 269338947495 T^{12} - 51421124475 p T^{13} + 10639939707 p^{2} T^{14} - 1660666050 p^{3} T^{15} + 183521187 p^{4} T^{16} - 14419053 p^{5} T^{17} + 1243256 p^{6} T^{18} - 268503 p^{7} T^{19} + 60429 p^{8} T^{20} - 8811 p^{9} T^{21} + 798 p^{10} T^{22} - 42 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( 1 + 15 T + 222 T^{2} + 2178 T^{3} + 19659 T^{4} + 180069 T^{5} + 1356134 T^{6} + 12927438 T^{7} + 95622579 T^{8} + 746933994 T^{9} + 5322701619 T^{10} + 32959233792 T^{11} + 285386764461 T^{12} + 32959233792 p T^{13} + 5322701619 p^{2} T^{14} + 746933994 p^{3} T^{15} + 95622579 p^{4} T^{16} + 12927438 p^{5} T^{17} + 1356134 p^{6} T^{18} + 180069 p^{7} T^{19} + 19659 p^{8} T^{20} + 2178 p^{9} T^{21} + 222 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 3 T - 156 T^{2} - 692 T^{3} + 4029 T^{4} + 93009 T^{5} + 751906 T^{6} - 5464728 T^{7} - 61999335 T^{8} + 7175122 T^{9} + 1232143473 T^{10} + 6354671856 T^{11} + 26893100905 T^{12} + 6354671856 p T^{13} + 1232143473 p^{2} T^{14} + 7175122 p^{3} T^{15} - 61999335 p^{4} T^{16} - 5464728 p^{5} T^{17} + 751906 p^{6} T^{18} + 93009 p^{7} T^{19} + 4029 p^{8} T^{20} - 692 p^{9} T^{21} - 156 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 3 T - 84 T^{2} - 350 T^{3} - 48 T^{4} + 80733 T^{5} + 420418 T^{6} - 4370418 T^{7} - 8932968 T^{8} - 113915288 T^{9} + 913697022 T^{10} + 14070237666 T^{11} - 104802324797 T^{12} + 14070237666 p T^{13} + 913697022 p^{2} T^{14} - 113915288 p^{3} T^{15} - 8932968 p^{4} T^{16} - 4370418 p^{5} T^{17} + 420418 p^{6} T^{18} + 80733 p^{7} T^{19} - 48 p^{8} T^{20} - 350 p^{9} T^{21} - 84 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 246 T^{2} - 1728 T^{3} + 33429 T^{4} + 375840 T^{5} - 1694146 T^{6} - 45344448 T^{7} - 63327222 T^{8} + 3008391840 T^{9} + 20386549170 T^{10} - 89267108832 T^{11} - 1881195165507 T^{12} - 89267108832 p T^{13} + 20386549170 p^{2} T^{14} + 3008391840 p^{3} T^{15} - 63327222 p^{4} T^{16} - 45344448 p^{5} T^{17} - 1694146 p^{6} T^{18} + 375840 p^{7} T^{19} + 33429 p^{8} T^{20} - 1728 p^{9} T^{21} - 246 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 + 12 T - 138 T^{2} - 1268 T^{3} + 17070 T^{4} + 53598 T^{5} - 1304234 T^{6} + 3526290 T^{7} + 68719068 T^{8} - 470908568 T^{9} - 1402775052 T^{10} + 24138453786 T^{11} + 101408995303 T^{12} + 24138453786 p T^{13} - 1402775052 p^{2} T^{14} - 470908568 p^{3} T^{15} + 68719068 p^{4} T^{16} + 3526290 p^{5} T^{17} - 1304234 p^{6} T^{18} + 53598 p^{7} T^{19} + 17070 p^{8} T^{20} - 1268 p^{9} T^{21} - 138 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 33 T + 708 T^{2} - 11996 T^{3} + 174345 T^{4} - 2233077 T^{5} + 25972498 T^{6} - 276952050 T^{7} + 2753048853 T^{8} - 25982259752 T^{9} + 235938467415 T^{10} - 2110605773970 T^{11} + 18780468373159 T^{12} - 2110605773970 p T^{13} + 235938467415 p^{2} T^{14} - 25982259752 p^{3} T^{15} + 2753048853 p^{4} T^{16} - 276952050 p^{5} T^{17} + 25972498 p^{6} T^{18} - 2233077 p^{7} T^{19} + 174345 p^{8} T^{20} - 11996 p^{9} T^{21} + 708 p^{10} T^{22} - 33 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 33 T + 510 T^{2} - 4446 T^{3} + 16671 T^{4} + 141069 T^{5} - 3124984 T^{6} + 27288360 T^{7} - 83223297 T^{8} - 882170946 T^{9} + 12432699873 T^{10} - 71516217942 T^{11} + 362393185551 T^{12} - 71516217942 p T^{13} + 12432699873 p^{2} T^{14} - 882170946 p^{3} T^{15} - 83223297 p^{4} T^{16} + 27288360 p^{5} T^{17} - 3124984 p^{6} T^{18} + 141069 p^{7} T^{19} + 16671 p^{8} T^{20} - 4446 p^{9} T^{21} + 510 p^{10} T^{22} - 33 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 9 T - 228 T^{2} + 3231 T^{3} + 19317 T^{4} - 508608 T^{5} - 47794 T^{6} + 51499179 T^{7} - 214197093 T^{8} - 3933968040 T^{9} + 40875889455 T^{10} + 148274963613 T^{11} - 4610618411295 T^{12} + 148274963613 p T^{13} + 40875889455 p^{2} T^{14} - 3933968040 p^{3} T^{15} - 214197093 p^{4} T^{16} + 51499179 p^{5} T^{17} - 47794 p^{6} T^{18} - 508608 p^{7} T^{19} + 19317 p^{8} T^{20} + 3231 p^{9} T^{21} - 228 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 15 T + 78 T^{2} + 1576 T^{3} - 42717 T^{4} + 517431 T^{5} - 1498652 T^{6} - 48622176 T^{7} + 951152553 T^{8} - 7859979524 T^{9} + 9374865825 T^{10} + 770633040408 T^{11} - 11486453899439 T^{12} + 770633040408 p T^{13} + 9374865825 p^{2} T^{14} - 7859979524 p^{3} T^{15} + 951152553 p^{4} T^{16} - 48622176 p^{5} T^{17} - 1498652 p^{6} T^{18} + 517431 p^{7} T^{19} - 42717 p^{8} T^{20} + 1576 p^{9} T^{21} + 78 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.71728825421780746763513165962, −3.23880577498883384389646832375, −2.99640499258973248179294375490, −2.98110128860672980408828884439, −2.88369969911416657165521413241, −2.85560952542137306173491207601, −2.72743771537326474042167179693, −2.65814116663456856482631036333, −2.49824103714018130905329292631, −2.38866165982343432024543814065, −2.33527865993842520642012374121, −2.09502879219841906909182670020, −2.08242087981993995256167566743, −1.81318603691250513889698598825, −1.73704447406699338304554622183, −1.60562938697263097818521600661, −1.44729158757537163438637765762, −1.39821524699995410657806521834, −1.28268480527986092238839871168, −1.05093545489228402052938234353, −0.936400010324329930953082964804, −0.76344460732520035621483390127, −0.73145251538387940281589159058, −0.32192466494411410996763868704, −0.30044551073686893832044183833, 0.30044551073686893832044183833, 0.32192466494411410996763868704, 0.73145251538387940281589159058, 0.76344460732520035621483390127, 0.936400010324329930953082964804, 1.05093545489228402052938234353, 1.28268480527986092238839871168, 1.39821524699995410657806521834, 1.44729158757537163438637765762, 1.60562938697263097818521600661, 1.73704447406699338304554622183, 1.81318603691250513889698598825, 2.08242087981993995256167566743, 2.09502879219841906909182670020, 2.33527865993842520642012374121, 2.38866165982343432024543814065, 2.49824103714018130905329292631, 2.65814116663456856482631036333, 2.72743771537326474042167179693, 2.85560952542137306173491207601, 2.88369969911416657165521413241, 2.98110128860672980408828884439, 2.99640499258973248179294375490, 3.23880577498883384389646832375, 3.71728825421780746763513165962
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.