Dirichlet series
| L(s) = 1 | + 6·2-s + 21·4-s − 6·5-s − 3·7-s + 54·8-s − 36·10-s − 12·11-s − 3·13-s − 18·14-s + 105·16-s + 9·17-s − 12·19-s − 126·20-s − 72·22-s − 6·23-s + 21·25-s − 18·26-s − 63·28-s − 12·29-s + 6·31-s + 150·32-s + 54·34-s + 18·35-s − 3·37-s − 72·38-s − 324·40-s − 24·41-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 21/2·4-s − 2.68·5-s − 1.13·7-s + 19.0·8-s − 11.3·10-s − 3.61·11-s − 0.832·13-s − 4.81·14-s + 26.2·16-s + 2.18·17-s − 2.75·19-s − 28.1·20-s − 15.3·22-s − 1.25·23-s + 21/5·25-s − 3.53·26-s − 11.9·28-s − 2.22·29-s + 1.07·31-s + 26.5·32-s + 9.26·34-s + 3.04·35-s − 0.493·37-s − 11.6·38-s − 51.2·40-s − 3.74·41-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 | \( 1 - 3 p T + 15 T^{2} - 9 p T^{3} + 3 p^{2} T^{4} - 3 p^{3} T^{5} + 59 T^{6} - 9 p^{2} T^{7} - 9 p^{3} T^{8} + 27 p^{2} T^{9} - 9 p^{3} T^{10} + 315 T^{11} - 741 T^{12} + 315 p T^{13} - 9 p^{5} T^{14} + 27 p^{5} T^{15} - 9 p^{7} T^{16} - 9 p^{7} T^{17} + 59 p^{6} T^{18} - 3 p^{10} T^{19} + 3 p^{10} T^{20} - 9 p^{10} T^{21} + 15 p^{10} T^{22} - 3 p^{12} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 + 6 T + 3 p T^{2} + 36 T^{3} + 111 T^{4} + 57 p T^{5} + 788 T^{6} + 2196 T^{7} + 4761 T^{8} + 10152 T^{9} + 26388 T^{10} + 67293 T^{11} + 155661 T^{12} + 67293 p T^{13} + 26388 p^{2} T^{14} + 10152 p^{3} T^{15} + 4761 p^{4} T^{16} + 2196 p^{5} T^{17} + 788 p^{6} T^{18} + 57 p^{8} T^{19} + 111 p^{8} T^{20} + 36 p^{9} T^{21} + 3 p^{11} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 + 3 T - 12 T^{2} - 44 T^{3} + 15 T^{4} + 129 T^{5} + 550 T^{6} + 2556 T^{7} - 2691 T^{8} - 30962 T^{9} - 15513 T^{10} + 105414 T^{11} + 217681 T^{12} + 105414 p T^{13} - 15513 p^{2} T^{14} - 30962 p^{3} T^{15} - 2691 p^{4} T^{16} + 2556 p^{5} T^{17} + 550 p^{6} T^{18} + 129 p^{7} T^{19} + 15 p^{8} T^{20} - 44 p^{9} T^{21} - 12 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 12 T + 87 T^{2} + 513 T^{3} + 2577 T^{4} + 11469 T^{5} + 48479 T^{6} + 198117 T^{7} + 773604 T^{8} + 2889108 T^{9} + 10427886 T^{10} + 36000711 T^{11} + 120262668 T^{12} + 36000711 p T^{13} + 10427886 p^{2} T^{14} + 2889108 p^{3} T^{15} + 773604 p^{4} T^{16} + 198117 p^{5} T^{17} + 48479 p^{6} T^{18} + 11469 p^{7} T^{19} + 2577 p^{8} T^{20} + 513 p^{9} T^{21} + 87 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 3 T - 3 p T^{2} - 98 T^{3} + 753 T^{4} + 1461 T^{5} - 9071 T^{6} - 11106 T^{7} + 58302 T^{8} + 25738 T^{9} + 274980 T^{10} + 19122 T^{11} - 10692236 T^{12} + 19122 p T^{13} + 274980 p^{2} T^{14} + 25738 p^{3} T^{15} + 58302 p^{4} T^{16} - 11106 p^{5} T^{17} - 9071 p^{6} T^{18} + 1461 p^{7} T^{19} + 753 p^{8} T^{20} - 98 p^{9} T^{21} - 3 p^{11} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( 1 + 6 T + 51 T^{2} + 387 T^{3} + 3603 T^{4} + 20427 T^{5} + 6193 p T^{6} + 834741 T^{7} + 5085072 T^{8} + 26501256 T^{9} + 153181206 T^{10} + 31848651 p T^{11} + 3662073816 T^{12} + 31848651 p^{2} T^{13} + 153181206 p^{2} T^{14} + 26501256 p^{3} T^{15} + 5085072 p^{4} T^{16} + 834741 p^{5} T^{17} + 6193 p^{7} T^{18} + 20427 p^{7} T^{19} + 3603 p^{8} T^{20} + 387 p^{9} T^{21} + 51 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 12 T + 96 T^{2} + 504 T^{3} + 1884 T^{4} + 5502 T^{5} + 7682 T^{6} + 11871 T^{7} + 213912 T^{8} + 2719602 T^{9} + 31068675 T^{10} + 269166771 T^{11} + 1683061413 T^{12} + 269166771 p T^{13} + 31068675 p^{2} T^{14} + 2719602 p^{3} T^{15} + 213912 p^{4} T^{16} + 11871 p^{5} T^{17} + 7682 p^{6} T^{18} + 5502 p^{7} T^{19} + 1884 p^{8} T^{20} + 504 p^{9} T^{21} + 96 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 6 T + 60 T^{2} - 530 T^{3} + 3552 T^{4} - 22956 T^{5} + 150634 T^{6} - 644634 T^{7} + 2944116 T^{8} - 16821776 T^{9} + 42941100 T^{10} - 121158966 T^{11} + 824278579 T^{12} - 121158966 p T^{13} + 42941100 p^{2} T^{14} - 16821776 p^{3} T^{15} + 2944116 p^{4} T^{16} - 644634 p^{5} T^{17} + 150634 p^{6} T^{18} - 22956 p^{7} T^{19} + 3552 p^{8} T^{20} - 530 p^{9} T^{21} + 60 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( 1 + 24 T + 276 T^{2} + 2061 T^{3} + 10911 T^{4} + 40101 T^{5} + 128714 T^{6} + 1367991 T^{7} + 19004949 T^{8} + 4277016 p T^{9} + 1214235063 T^{10} + 7381493937 T^{11} + 45314985201 T^{12} + 7381493937 p T^{13} + 1214235063 p^{2} T^{14} + 4277016 p^{4} T^{15} + 19004949 p^{4} T^{16} + 1367991 p^{5} T^{17} + 128714 p^{6} T^{18} + 40101 p^{7} T^{19} + 10911 p^{8} T^{20} + 2061 p^{9} T^{21} + 276 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 6 T + 24 T^{2} - 242 T^{3} + 3048 T^{4} - 18240 T^{5} + 107686 T^{6} - 684054 T^{7} + 5981724 T^{8} - 60993776 T^{9} + 316582968 T^{10} - 2264791602 T^{11} + 15738134299 T^{12} - 2264791602 p T^{13} + 316582968 p^{2} T^{14} - 60993776 p^{3} T^{15} + 5981724 p^{4} T^{16} - 684054 p^{5} T^{17} + 107686 p^{6} T^{18} - 18240 p^{7} T^{19} + 3048 p^{8} T^{20} - 242 p^{9} T^{21} + 24 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 21 T + 177 T^{2} + 639 T^{3} - 1500 T^{4} - 41820 T^{5} - 309136 T^{6} + 104535 T^{7} + 23189445 T^{8} + 203244822 T^{9} + 692444061 T^{10} - 1294793487 T^{11} - 25455528993 T^{12} - 1294793487 p T^{13} + 692444061 p^{2} T^{14} + 203244822 p^{3} T^{15} + 23189445 p^{4} T^{16} + 104535 p^{5} T^{17} - 309136 p^{6} T^{18} - 41820 p^{7} T^{19} - 1500 p^{8} T^{20} + 639 p^{9} T^{21} + 177 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( 1 + 15 T + 78 T^{2} + 18 T^{3} + 3099 T^{4} + 79107 T^{5} + 445640 T^{6} - 2803374 T^{7} - 33176943 T^{8} - 4607118 T^{9} + 750424779 T^{10} - 14514804090 T^{11} - 224732447481 T^{12} - 14514804090 p T^{13} + 750424779 p^{2} T^{14} - 4607118 p^{3} T^{15} - 33176943 p^{4} T^{16} - 2803374 p^{5} T^{17} + 445640 p^{6} T^{18} + 79107 p^{7} T^{19} + 3099 p^{8} T^{20} + 18 p^{9} T^{21} + 78 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 33 T + 600 T^{2} - 8036 T^{3} + 92913 T^{4} - 1026501 T^{5} + 11002780 T^{6} - 111633966 T^{7} + 1067991759 T^{8} - 9660914408 T^{9} + 83400019635 T^{10} - 689079109476 T^{11} + 5475797629045 T^{12} - 689079109476 p T^{13} + 83400019635 p^{2} T^{14} - 9660914408 p^{3} T^{15} + 1067991759 p^{4} T^{16} - 111633966 p^{5} T^{17} + 11002780 p^{6} T^{18} - 1026501 p^{7} T^{19} + 92913 p^{8} T^{20} - 8036 p^{9} T^{21} + 600 p^{10} T^{22} - 33 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 42 T + 924 T^{2} - 14201 T^{3} + 172221 T^{4} - 1807881 T^{5} + 17756848 T^{6} - 172138131 T^{7} + 1642732227 T^{8} - 14821179572 T^{9} + 1850441097 p T^{10} - 987365131869 T^{11} + 7925147731483 T^{12} - 987365131869 p T^{13} + 1850441097 p^{3} T^{14} - 14821179572 p^{3} T^{15} + 1642732227 p^{4} T^{16} - 172138131 p^{5} T^{17} + 17756848 p^{6} T^{18} - 1807881 p^{7} T^{19} + 172221 p^{8} T^{20} - 14201 p^{9} T^{21} + 924 p^{10} T^{22} - 42 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( 1 + 48 T + 915 T^{2} + 6445 T^{3} - 63129 T^{4} - 1950603 T^{5} - 19135835 T^{6} - 26231697 T^{7} + 1657331316 T^{8} + 21434758414 T^{9} + 82959754584 T^{10} - 965251960509 T^{11} - 16099610505752 T^{12} - 965251960509 p T^{13} + 82959754584 p^{2} T^{14} + 21434758414 p^{3} T^{15} + 1657331316 p^{4} T^{16} - 26231697 p^{5} T^{17} - 19135835 p^{6} T^{18} - 1950603 p^{7} T^{19} - 63129 p^{8} T^{20} + 6445 p^{9} T^{21} + 915 p^{10} T^{22} + 48 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 12 T + 87 T^{2} + 873 T^{3} + 4737 T^{4} - 55329 T^{5} - 663637 T^{6} - 3237687 T^{7} - 54357750 T^{8} - 213032268 T^{9} + 6408468522 T^{10} + 85292792685 T^{11} + 711247470576 T^{12} + 85292792685 p T^{13} + 6408468522 p^{2} T^{14} - 213032268 p^{3} T^{15} - 54357750 p^{4} T^{16} - 3237687 p^{5} T^{17} - 663637 p^{6} T^{18} - 55329 p^{7} T^{19} + 4737 p^{8} T^{20} + 873 p^{9} T^{21} + 87 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( 1 + 3 T + 132 T^{2} + 2008 T^{3} + 25377 T^{4} + 125193 T^{5} + 3962692 T^{6} + 15909768 T^{7} + 250495857 T^{8} + 1659509836 T^{9} + 21828189837 T^{10} - 4230491340 T^{11} + 2739349132513 T^{12} - 4230491340 p T^{13} + 21828189837 p^{2} T^{14} + 1659509836 p^{3} T^{15} + 250495857 p^{4} T^{16} + 15909768 p^{5} T^{17} + 3962692 p^{6} T^{18} + 125193 p^{7} T^{19} + 25377 p^{8} T^{20} + 2008 p^{9} T^{21} + 132 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.35336836103518415788280650006, −3.33770436523774534867032863471, −3.27798184517242948414552330028, −3.24930917477564873574310466927, −3.24627131995384667517310158913, −3.01197878052756478087064340736, −2.70015059471794645954768134789, −2.67001699657231865237074776980, −2.60933967613701859269600672650, −2.59286147251713898150236462377, −2.44720108802644517258148956026, −2.35406609192328040647337958184, −2.28075391677472717013559684606, −2.20666869344269542209821326743, −2.09329264537783520884820855872, −1.85413185377550065953813161562, −1.84586039837055695714881246015, −1.64136041999305248079609881916, −1.44374389400836513838790960303, −1.37531704628227921978595179076, −1.12521454920176276856875776862, −0.74309652909208492248116972486, −0.38624318065652344274110911486, −0.28439879739354273438849598911, −0.11188077400779826054742399862, 0.11188077400779826054742399862, 0.28439879739354273438849598911, 0.38624318065652344274110911486, 0.74309652909208492248116972486, 1.12521454920176276856875776862, 1.37531704628227921978595179076, 1.44374389400836513838790960303, 1.64136041999305248079609881916, 1.84586039837055695714881246015, 1.85413185377550065953813161562, 2.09329264537783520884820855872, 2.20666869344269542209821326743, 2.28075391677472717013559684606, 2.35406609192328040647337958184, 2.44720108802644517258148956026, 2.59286147251713898150236462377, 2.60933967613701859269600672650, 2.67001699657231865237074776980, 2.70015059471794645954768134789, 3.01197878052756478087064340736, 3.24627131995384667517310158913, 3.24930917477564873574310466927, 3.27798184517242948414552330028, 3.33770436523774534867032863471, 3.35336836103518415788280650006
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.