Dirichlet series
| L(s) = 1 | − 4·2-s − 8·3-s + 2·4-s + 12·5-s + 32·6-s − 16·7-s + 12·8-s + 20·9-s − 48·10-s − 16·11-s − 16·12-s − 8·13-s + 64·14-s − 96·15-s − 17·16-s − 80·18-s + 24·20-s + 128·21-s + 64·22-s − 16·23-s − 96·24-s + 78·25-s + 32·26-s + 8·27-s − 32·28-s − 16·29-s + 384·30-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 4.61·3-s + 4-s + 5.36·5-s + 13.0·6-s − 6.04·7-s + 4.24·8-s + 20/3·9-s − 15.1·10-s − 4.82·11-s − 4.61·12-s − 2.21·13-s + 17.1·14-s − 24.7·15-s − 4.25·16-s − 18.8·18-s + 5.36·20-s + 27.9·21-s + 13.6·22-s − 3.33·23-s − 19.5·24-s + 78/5·25-s + 6.27·26-s + 1.53·27-s − 6.04·28-s − 2.97·29-s + 70.1·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 17^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 17^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(5^{12} \cdot 17^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.56851\times 10^{12}\) |
| Root analytic conductor: | \(3.39681\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 5^{12} \cdot 17^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 - T )^{12} \) |
| 17 | \( 1 \) | |
| good | 2 | \( 1 + p^{2} T + 7 p T^{2} + 9 p^{2} T^{3} + 85 T^{4} + 11 p^{4} T^{5} + 85 p^{2} T^{6} + 77 p^{3} T^{7} + 1063 T^{8} + 435 p^{2} T^{9} + 1371 p T^{10} + 1027 p^{2} T^{11} + 5969 T^{12} + 1027 p^{3} T^{13} + 1371 p^{3} T^{14} + 435 p^{5} T^{15} + 1063 p^{4} T^{16} + 77 p^{8} T^{17} + 85 p^{8} T^{18} + 11 p^{11} T^{19} + 85 p^{8} T^{20} + 9 p^{11} T^{21} + 7 p^{11} T^{22} + p^{13} T^{23} + p^{12} T^{24} \) |
| 3 | \( 1 + 8 T + 44 T^{2} + 184 T^{3} + 8 p^{4} T^{4} + 1976 T^{5} + 5396 T^{6} + 13360 T^{7} + 30529 T^{8} + 64808 T^{9} + 129212 T^{10} + 242728 T^{11} + 432086 T^{12} + 242728 p T^{13} + 129212 p^{2} T^{14} + 64808 p^{3} T^{15} + 30529 p^{4} T^{16} + 13360 p^{5} T^{17} + 5396 p^{6} T^{18} + 1976 p^{7} T^{19} + 8 p^{12} T^{20} + 184 p^{9} T^{21} + 44 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 + 16 T + 160 T^{2} + 1160 T^{3} + 982 p T^{4} + 34536 T^{5} + 153164 T^{6} + 609048 T^{7} + 2214977 T^{8} + 1060128 p T^{9} + 23153912 T^{10} + 67428528 T^{11} + 184297858 T^{12} + 67428528 p T^{13} + 23153912 p^{2} T^{14} + 1060128 p^{4} T^{15} + 2214977 p^{4} T^{16} + 609048 p^{5} T^{17} + 153164 p^{6} T^{18} + 34536 p^{7} T^{19} + 982 p^{9} T^{20} + 1160 p^{9} T^{21} + 160 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 16 T + 192 T^{2} + 1704 T^{3} + 12928 T^{4} + 83992 T^{5} + 488196 T^{6} + 2544384 T^{7} + 1101805 p T^{8} + 52807504 T^{9} + 212547564 T^{10} + 789633344 T^{11} + 2722360418 T^{12} + 789633344 p T^{13} + 212547564 p^{2} T^{14} + 52807504 p^{3} T^{15} + 1101805 p^{5} T^{16} + 2544384 p^{5} T^{17} + 488196 p^{6} T^{18} + 83992 p^{7} T^{19} + 12928 p^{8} T^{20} + 1704 p^{9} T^{21} + 192 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 8 T + 96 T^{2} + 48 p T^{3} + 4468 T^{4} + 24376 T^{5} + 135828 T^{6} + 640576 T^{7} + 3043711 T^{8} + 12754792 T^{9} + 53584828 T^{10} + 203115752 T^{11} + 767649052 T^{12} + 203115752 p T^{13} + 53584828 p^{2} T^{14} + 12754792 p^{3} T^{15} + 3043711 p^{4} T^{16} + 640576 p^{5} T^{17} + 135828 p^{6} T^{18} + 24376 p^{7} T^{19} + 4468 p^{8} T^{20} + 48 p^{10} T^{21} + 96 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 136 T^{2} + 16 T^{3} + 8694 T^{4} + 112 T^{5} + 350152 T^{6} - 98432 T^{7} + 10144191 T^{8} - 6779904 T^{9} + 233582832 T^{10} - 222178912 T^{11} + 4668809460 T^{12} - 222178912 p T^{13} + 233582832 p^{2} T^{14} - 6779904 p^{3} T^{15} + 10144191 p^{4} T^{16} - 98432 p^{5} T^{17} + 350152 p^{6} T^{18} + 112 p^{7} T^{19} + 8694 p^{8} T^{20} + 16 p^{9} T^{21} + 136 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( 1 + 16 T + 288 T^{2} + 3160 T^{3} + 34024 T^{4} + 288408 T^{5} + 2336672 T^{6} + 16287808 T^{7} + 108182549 T^{8} + 644375808 T^{9} + 3668311868 T^{10} + 19118203032 T^{11} + 95514424750 T^{12} + 19118203032 p T^{13} + 3668311868 p^{2} T^{14} + 644375808 p^{3} T^{15} + 108182549 p^{4} T^{16} + 16287808 p^{5} T^{17} + 2336672 p^{6} T^{18} + 288408 p^{7} T^{19} + 34024 p^{8} T^{20} + 3160 p^{9} T^{21} + 288 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 16 T + 344 T^{2} + 3744 T^{3} + 46776 T^{4} + 388992 T^{5} + 3605672 T^{6} + 24458352 T^{7} + 186387123 T^{8} + 1083675456 T^{9} + 7238471232 T^{10} + 37619665952 T^{11} + 228960125136 T^{12} + 37619665952 p T^{13} + 7238471232 p^{2} T^{14} + 1083675456 p^{3} T^{15} + 186387123 p^{4} T^{16} + 24458352 p^{5} T^{17} + 3605672 p^{6} T^{18} + 388992 p^{7} T^{19} + 46776 p^{8} T^{20} + 3744 p^{9} T^{21} + 344 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 24 T + 540 T^{2} + 8032 T^{3} + 3518 p T^{4} + 1204576 T^{5} + 12238640 T^{6} + 108231320 T^{7} + 887813751 T^{8} + 6528377840 T^{9} + 44782669908 T^{10} + 279085389568 T^{11} + 1627060275182 T^{12} + 279085389568 p T^{13} + 44782669908 p^{2} T^{14} + 6528377840 p^{3} T^{15} + 887813751 p^{4} T^{16} + 108231320 p^{5} T^{17} + 12238640 p^{6} T^{18} + 1204576 p^{7} T^{19} + 3518 p^{9} T^{20} + 8032 p^{9} T^{21} + 540 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 24 T + 544 T^{2} + 8464 T^{3} + 119152 T^{4} + 1400048 T^{5} + 15042760 T^{6} + 143029432 T^{7} + 1257115355 T^{8} + 10017377104 T^{9} + 74284445768 T^{10} + 504849591712 T^{11} + 3202683725104 T^{12} + 504849591712 p T^{13} + 74284445768 p^{2} T^{14} + 10017377104 p^{3} T^{15} + 1257115355 p^{4} T^{16} + 143029432 p^{5} T^{17} + 15042760 p^{6} T^{18} + 1400048 p^{7} T^{19} + 119152 p^{8} T^{20} + 8464 p^{9} T^{21} + 544 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 8 T + 340 T^{2} + 2616 T^{3} + 53826 T^{4} + 400024 T^{5} + 5302052 T^{6} + 38308776 T^{7} + 369886863 T^{8} + 2605498768 T^{9} + 19954283976 T^{10} + 135146134640 T^{11} + 888619861532 T^{12} + 135146134640 p T^{13} + 19954283976 p^{2} T^{14} + 2605498768 p^{3} T^{15} + 369886863 p^{4} T^{16} + 38308776 p^{5} T^{17} + 5302052 p^{6} T^{18} + 400024 p^{7} T^{19} + 53826 p^{8} T^{20} + 2616 p^{9} T^{21} + 340 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 16 T + 444 T^{2} + 5672 T^{3} + 90628 T^{4} + 961624 T^{5} + 11442656 T^{6} + 103526016 T^{7} + 1005460955 T^{8} + 183434976 p T^{9} + 65159925036 T^{10} + 446959697392 T^{11} + 3203324083092 T^{12} + 446959697392 p T^{13} + 65159925036 p^{2} T^{14} + 183434976 p^{4} T^{15} + 1005460955 p^{4} T^{16} + 103526016 p^{5} T^{17} + 11442656 p^{6} T^{18} + 961624 p^{7} T^{19} + 90628 p^{8} T^{20} + 5672 p^{9} T^{21} + 444 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 32 T + 876 T^{2} + 16384 T^{3} + 5784 p T^{4} + 3720104 T^{5} + 46291536 T^{6} + 505061192 T^{7} + 5083475999 T^{8} + 46073394168 T^{9} + 388312839988 T^{10} + 2982765007224 T^{11} + 454938276012 p T^{12} + 2982765007224 p T^{13} + 388312839988 p^{2} T^{14} + 46073394168 p^{3} T^{15} + 5083475999 p^{4} T^{16} + 505061192 p^{5} T^{17} + 46291536 p^{6} T^{18} + 3720104 p^{7} T^{19} + 5784 p^{9} T^{20} + 16384 p^{9} T^{21} + 876 p^{10} T^{22} + 32 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 296 T^{2} + 552 T^{3} + 39766 T^{4} + 159672 T^{5} + 3504240 T^{6} + 20930448 T^{7} + 258103207 T^{8} + 1714170704 T^{9} + 17518104008 T^{10} + 105811189536 T^{11} + 1025985617076 T^{12} + 105811189536 p T^{13} + 17518104008 p^{2} T^{14} + 1714170704 p^{3} T^{15} + 258103207 p^{4} T^{16} + 20930448 p^{5} T^{17} + 3504240 p^{6} T^{18} + 159672 p^{7} T^{19} + 39766 p^{8} T^{20} + 552 p^{9} T^{21} + 296 p^{10} T^{22} + p^{12} T^{24} \) | |
| 59 | \( 1 - 8 T + 444 T^{2} - 3368 T^{3} + 99154 T^{4} - 702104 T^{5} + 14634220 T^{6} - 95852536 T^{7} + 1586255871 T^{8} - 9522535632 T^{9} + 132763969272 T^{10} - 722084927504 T^{11} + 8781576906172 T^{12} - 722084927504 p T^{13} + 132763969272 p^{2} T^{14} - 9522535632 p^{3} T^{15} + 1586255871 p^{4} T^{16} - 95852536 p^{5} T^{17} + 14634220 p^{6} T^{18} - 702104 p^{7} T^{19} + 99154 p^{8} T^{20} - 3368 p^{9} T^{21} + 444 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 24 T + 700 T^{2} + 11576 T^{3} + 201290 T^{4} + 2578616 T^{5} + 33685596 T^{6} + 355678424 T^{7} + 3821265519 T^{8} + 34709774960 T^{9} + 323492221768 T^{10} + 2612061415984 T^{11} + 21825496496076 T^{12} + 2612061415984 p T^{13} + 323492221768 p^{2} T^{14} + 34709774960 p^{3} T^{15} + 3821265519 p^{4} T^{16} + 355678424 p^{5} T^{17} + 33685596 p^{6} T^{18} + 2578616 p^{7} T^{19} + 201290 p^{8} T^{20} + 11576 p^{9} T^{21} + 700 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 8 T + 504 T^{2} + 3520 T^{3} + 125076 T^{4} + 771528 T^{5} + 20245948 T^{6} + 110768576 T^{7} + 2396805935 T^{8} + 11699526984 T^{9} + 220875304092 T^{10} + 968880918232 T^{11} + 16395435908988 T^{12} + 968880918232 p T^{13} + 220875304092 p^{2} T^{14} + 11699526984 p^{3} T^{15} + 2396805935 p^{4} T^{16} + 110768576 p^{5} T^{17} + 20245948 p^{6} T^{18} + 771528 p^{7} T^{19} + 125076 p^{8} T^{20} + 3520 p^{9} T^{21} + 504 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 16 T + 808 T^{2} - 10296 T^{3} + 289596 T^{4} - 3056984 T^{5} + 62456068 T^{6} - 560135232 T^{7} + 9200146635 T^{8} - 71287446640 T^{9} + 989297992700 T^{10} - 6687718659552 T^{11} + 80373057726690 T^{12} - 6687718659552 p T^{13} + 989297992700 p^{2} T^{14} - 71287446640 p^{3} T^{15} + 9200146635 p^{4} T^{16} - 560135232 p^{5} T^{17} + 62456068 p^{6} T^{18} - 3056984 p^{7} T^{19} + 289596 p^{8} T^{20} - 10296 p^{9} T^{21} + 808 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 16 T + 696 T^{2} + 10416 T^{3} + 237390 T^{4} + 3197744 T^{5} + 51681800 T^{6} + 613165296 T^{7} + 7903591679 T^{8} + 81803324608 T^{9} + 887157068672 T^{10} + 7987730056672 T^{11} + 74598795265604 T^{12} + 7987730056672 p T^{13} + 887157068672 p^{2} T^{14} + 81803324608 p^{3} T^{15} + 7903591679 p^{4} T^{16} + 613165296 p^{5} T^{17} + 51681800 p^{6} T^{18} + 3197744 p^{7} T^{19} + 237390 p^{8} T^{20} + 10416 p^{9} T^{21} + 696 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 40 T + 1260 T^{2} + 28128 T^{3} + 548558 T^{4} + 9033984 T^{5} + 135362672 T^{6} + 1810938328 T^{7} + 22468082291 T^{8} + 254893959312 T^{9} + 2707939576316 T^{10} + 26586919047408 T^{11} + 245445099063342 T^{12} + 26586919047408 p T^{13} + 2707939576316 p^{2} T^{14} + 254893959312 p^{3} T^{15} + 22468082291 p^{4} T^{16} + 1810938328 p^{5} T^{17} + 135362672 p^{6} T^{18} + 9033984 p^{7} T^{19} + 548558 p^{8} T^{20} + 28128 p^{9} T^{21} + 1260 p^{10} T^{22} + 40 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 40 T + 1160 T^{2} + 25208 T^{3} + 466372 T^{4} + 7446848 T^{5} + 107046500 T^{6} + 1392472240 T^{7} + 16754415935 T^{8} + 187037621768 T^{9} + 1963434748068 T^{10} + 19393159406024 T^{11} + 181718580204412 T^{12} + 19393159406024 p T^{13} + 1963434748068 p^{2} T^{14} + 187037621768 p^{3} T^{15} + 16754415935 p^{4} T^{16} + 1392472240 p^{5} T^{17} + 107046500 p^{6} T^{18} + 7446848 p^{7} T^{19} + 466372 p^{8} T^{20} + 25208 p^{9} T^{21} + 1160 p^{10} T^{22} + 40 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 8 T + 892 T^{2} - 6056 T^{3} + 367562 T^{4} - 2102776 T^{5} + 93376936 T^{6} - 448799464 T^{7} + 16487713063 T^{8} - 67003569168 T^{9} + 2160668806220 T^{10} - 7563686935040 T^{11} + 217901060041728 T^{12} - 7563686935040 p T^{13} + 2160668806220 p^{2} T^{14} - 67003569168 p^{3} T^{15} + 16487713063 p^{4} T^{16} - 448799464 p^{5} T^{17} + 93376936 p^{6} T^{18} - 2102776 p^{7} T^{19} + 367562 p^{8} T^{20} - 6056 p^{9} T^{21} + 892 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 32 T + 1032 T^{2} + 22696 T^{3} + 464384 T^{4} + 7992344 T^{5} + 1315776 p T^{6} + 1831437904 T^{7} + 24500975355 T^{8} + 302835242448 T^{9} + 3505572177864 T^{10} + 37958649645216 T^{11} + 385822241517232 T^{12} + 37958649645216 p T^{13} + 3505572177864 p^{2} T^{14} + 302835242448 p^{3} T^{15} + 24500975355 p^{4} T^{16} + 1831437904 p^{5} T^{17} + 1315776 p^{7} T^{18} + 7992344 p^{7} T^{19} + 464384 p^{8} T^{20} + 22696 p^{9} T^{21} + 1032 p^{10} T^{22} + 32 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.41335051165401718035626221423, −3.21679599713821880258174712364, −3.17716692716915539758908411966, −3.16373498657202430105749384750, −3.12978674148803160128635019129, −3.08130702326072392108494569746, −2.90621645228161039720524523040, −2.85122272085309536310483905797, −2.83961591924634940499078204360, −2.62873673497050936297863535353, −2.62583922440878053009276109246, −2.47571826375467861706263746244, −2.41771968299739893243985036794, −2.30607110524205906657832557871, −2.14404770043725092155692082593, −2.00618125835521516194538222340, −1.90703864321361850110089219134, −1.85226751210875602443544358340, −1.74122097314910038235995502145, −1.64934399612998701186618442407, −1.45623913074766883830594289825, −1.43614997498796148765797655920, −1.30443674283601494176442586575, −1.09646632346117290797616467005, −0.995391354972832696902853205265, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.995391354972832696902853205265, 1.09646632346117290797616467005, 1.30443674283601494176442586575, 1.43614997498796148765797655920, 1.45623913074766883830594289825, 1.64934399612998701186618442407, 1.74122097314910038235995502145, 1.85226751210875602443544358340, 1.90703864321361850110089219134, 2.00618125835521516194538222340, 2.14404770043725092155692082593, 2.30607110524205906657832557871, 2.41771968299739893243985036794, 2.47571826375467861706263746244, 2.62583922440878053009276109246, 2.62873673497050936297863535353, 2.83961591924634940499078204360, 2.85122272085309536310483905797, 2.90621645228161039720524523040, 3.08130702326072392108494569746, 3.12978674148803160128635019129, 3.16373498657202430105749384750, 3.17716692716915539758908411966, 3.21679599713821880258174712364, 3.41335051165401718035626221423
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.