Dirichlet series
| L(s) = 1 | − 3·2-s + 3·4-s + 12·5-s − 3·7-s − 36·10-s − 3·11-s + 6·13-s + 9·14-s + 6·16-s + 9·17-s − 12·19-s + 36·20-s + 9·22-s + 21·23-s + 75·25-s − 18·26-s − 9·28-s + 6·29-s + 6·31-s − 30·32-s − 27·34-s − 36·35-s − 3·37-s + 36·38-s − 15·41-s − 30·43-s − 9·44-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3/2·4-s + 5.36·5-s − 1.13·7-s − 11.3·10-s − 0.904·11-s + 1.66·13-s + 2.40·14-s + 3/2·16-s + 2.18·17-s − 2.75·19-s + 8.04·20-s + 1.91·22-s + 4.37·23-s + 15·25-s − 3.53·26-s − 1.70·28-s + 1.11·29-s + 1.07·31-s − 5.30·32-s − 4.63·34-s − 6.08·35-s − 0.493·37-s + 5.83·38-s − 2.34·41-s − 4.57·43-s − 1.35·44-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 | \( 1 + 3 T + 3 p T^{2} + 9 T^{3} + 3 T^{4} - 3 p T^{5} - 11 p T^{6} - 45 T^{7} - 9 p^{2} T^{8} - 27 T^{9} + 9 p^{3} T^{10} + 135 p T^{11} + 393 T^{12} + 135 p^{2} T^{13} + 9 p^{5} T^{14} - 27 p^{3} T^{15} - 9 p^{6} T^{16} - 45 p^{5} T^{17} - 11 p^{7} T^{18} - 3 p^{8} T^{19} + 3 p^{8} T^{20} + 9 p^{9} T^{21} + 3 p^{11} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 - 12 T + 69 T^{2} - 234 T^{3} + 444 T^{4} - 201 T^{5} - 161 p T^{6} - 1521 T^{7} + 22158 T^{8} - 3078 p^{2} T^{9} + 128214 T^{10} - 39582 T^{11} - 204114 T^{12} - 39582 p T^{13} + 128214 p^{2} T^{14} - 3078 p^{5} T^{15} + 22158 p^{4} T^{16} - 1521 p^{5} T^{17} - 161 p^{7} T^{18} - 201 p^{7} T^{19} + 444 p^{8} T^{20} - 234 p^{9} T^{21} + 69 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 + 3 T + 6 T^{2} + 10 T^{3} + 33 T^{4} - 33 T^{5} + 64 T^{6} + 396 T^{7} + 549 p T^{8} + 7918 T^{9} + 18507 T^{10} + 24000 T^{11} + 177235 T^{12} + 24000 p T^{13} + 18507 p^{2} T^{14} + 7918 p^{3} T^{15} + 549 p^{5} T^{16} + 396 p^{5} T^{17} + 64 p^{6} T^{18} - 33 p^{7} T^{19} + 33 p^{8} T^{20} + 10 p^{9} T^{21} + 6 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 3 T + 6 T^{2} + 54 T^{3} + 345 T^{4} + 993 T^{5} + 4766 T^{6} + 16344 T^{7} + 71937 T^{8} + 262710 T^{9} + 798399 T^{10} + 2750058 T^{11} + 12777477 T^{12} + 2750058 p T^{13} + 798399 p^{2} T^{14} + 262710 p^{3} T^{15} + 71937 p^{4} T^{16} + 16344 p^{5} T^{17} + 4766 p^{6} T^{18} + 993 p^{7} T^{19} + 345 p^{8} T^{20} + 54 p^{9} T^{21} + 6 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 6 T + 15 T^{2} + 10 T^{3} - 300 T^{4} + 66 p T^{5} + 82 T^{6} - 5274 T^{7} + 32139 T^{8} - 3098 T^{9} - 10563 p T^{10} + 64734 p T^{11} - 6128183 T^{12} + 64734 p^{2} T^{13} - 10563 p^{3} T^{14} - 3098 p^{3} T^{15} + 32139 p^{4} T^{16} - 5274 p^{5} T^{17} + 82 p^{6} T^{18} + 66 p^{8} T^{19} - 300 p^{8} T^{20} + 10 p^{9} T^{21} + 15 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( 1 - 21 T + 168 T^{2} - 288 T^{3} - 6117 T^{4} + 61323 T^{5} - 223870 T^{6} - 513414 T^{7} + 10295919 T^{8} - 52171398 T^{9} + 59309289 T^{10} + 941234742 T^{11} - 7207345695 T^{12} + 941234742 p T^{13} + 59309289 p^{2} T^{14} - 52171398 p^{3} T^{15} + 10295919 p^{4} T^{16} - 513414 p^{5} T^{17} - 223870 p^{6} T^{18} + 61323 p^{7} T^{19} - 6117 p^{8} T^{20} - 288 p^{9} T^{21} + 168 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 6 T - 21 T^{2} + 126 T^{3} + 471 T^{4} + 2847 T^{5} - 40432 T^{6} + 104418 T^{7} - 588357 T^{8} + 817344 T^{9} + 31802490 T^{10} - 153874539 T^{11} + 196961865 T^{12} - 153874539 p T^{13} + 31802490 p^{2} T^{14} + 817344 p^{3} T^{15} - 588357 p^{4} T^{16} + 104418 p^{5} T^{17} - 40432 p^{6} T^{18} + 2847 p^{7} T^{19} + 471 p^{8} T^{20} + 126 p^{9} T^{21} - 21 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 6 T + 24 T^{2} - 314 T^{3} + 3084 T^{4} - 7350 T^{5} + 38854 T^{6} - 540576 T^{7} + 1673172 T^{8} - 2486720 T^{9} + 68080044 T^{10} - 371144496 T^{11} + 182704687 T^{12} - 371144496 p T^{13} + 68080044 p^{2} T^{14} - 2486720 p^{3} T^{15} + 1673172 p^{4} T^{16} - 540576 p^{5} T^{17} + 38854 p^{6} T^{18} - 7350 p^{7} T^{19} + 3084 p^{8} T^{20} - 314 p^{9} T^{21} + 24 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( 1 + 15 T + 213 T^{2} + 2115 T^{3} + 19596 T^{4} + 157506 T^{5} + 1233878 T^{6} + 8928837 T^{7} + 66584925 T^{8} + 476310456 T^{9} + 3430520001 T^{10} + 23249431161 T^{11} + 153741756273 T^{12} + 23249431161 p T^{13} + 3430520001 p^{2} T^{14} + 476310456 p^{3} T^{15} + 66584925 p^{4} T^{16} + 8928837 p^{5} T^{17} + 1233878 p^{6} T^{18} + 157506 p^{7} T^{19} + 19596 p^{8} T^{20} + 2115 p^{9} T^{21} + 213 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 30 T + 384 T^{2} + 2350 T^{3} - 1668 T^{4} - 165030 T^{5} - 1351934 T^{6} - 2195892 T^{7} + 1224108 p T^{8} + 511565416 T^{9} + 1137795888 T^{10} - 17255158140 T^{11} - 191561429657 T^{12} - 17255158140 p T^{13} + 1137795888 p^{2} T^{14} + 511565416 p^{3} T^{15} + 1224108 p^{5} T^{16} - 2195892 p^{5} T^{17} - 1351934 p^{6} T^{18} - 165030 p^{7} T^{19} - 1668 p^{8} T^{20} + 2350 p^{9} T^{21} + 384 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 21 T + 348 T^{2} + 90 p T^{3} + 43968 T^{4} + 8565 p T^{5} + 3391160 T^{6} + 27268470 T^{7} + 216384498 T^{8} + 1695745908 T^{9} + 13136482818 T^{10} + 97262056422 T^{11} + 689007577155 T^{12} + 97262056422 p T^{13} + 13136482818 p^{2} T^{14} + 1695745908 p^{3} T^{15} + 216384498 p^{4} T^{16} + 27268470 p^{5} T^{17} + 3391160 p^{6} T^{18} + 8565 p^{8} T^{19} + 43968 p^{8} T^{20} + 90 p^{10} T^{21} + 348 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( 1 - 30 T + 375 T^{2} - 2277 T^{3} - 699 T^{4} + 121389 T^{5} - 569263 T^{6} - 3489057 T^{7} + 44231742 T^{8} - 69704010 T^{9} - 1160991180 T^{10} + 1776655107 T^{11} + 52093945536 T^{12} + 1776655107 p T^{13} - 1160991180 p^{2} T^{14} - 69704010 p^{3} T^{15} + 44231742 p^{4} T^{16} - 3489057 p^{5} T^{17} - 569263 p^{6} T^{18} + 121389 p^{7} T^{19} - 699 p^{8} T^{20} - 2277 p^{9} T^{21} + 375 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 30 T + 555 T^{2} + 7435 T^{3} + 82905 T^{4} + 797169 T^{5} + 6868945 T^{6} + 52027101 T^{7} + 332247780 T^{8} + 1686283252 T^{9} + 5903776218 T^{10} + 7573975233 T^{11} - 54933098744 T^{12} + 7573975233 p T^{13} + 5903776218 p^{2} T^{14} + 1686283252 p^{3} T^{15} + 332247780 p^{4} T^{16} + 52027101 p^{5} T^{17} + 6868945 p^{6} T^{18} + 797169 p^{7} T^{19} + 82905 p^{8} T^{20} + 7435 p^{9} T^{21} + 555 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 39 T + 807 T^{2} + 11611 T^{3} + 131766 T^{4} + 1246656 T^{5} + 9980848 T^{6} + 66946653 T^{7} + 378814707 T^{8} + 1943205748 T^{9} + 10439030541 T^{10} + 66587679033 T^{11} + 503739920563 T^{12} + 66587679033 p T^{13} + 10439030541 p^{2} T^{14} + 1943205748 p^{3} T^{15} + 378814707 p^{4} T^{16} + 66946653 p^{5} T^{17} + 9980848 p^{6} T^{18} + 1246656 p^{7} T^{19} + 131766 p^{8} T^{20} + 11611 p^{9} T^{21} + 807 p^{10} T^{22} + 39 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( 1 - 15 T + 186 T^{2} - 2276 T^{3} + 29913 T^{4} - 328911 T^{5} + 3426148 T^{6} - 34516026 T^{7} + 344384307 T^{8} - 3532313684 T^{9} + 35719673025 T^{10} - 324531893478 T^{11} + 2775733548577 T^{12} - 324531893478 p T^{13} + 35719673025 p^{2} T^{14} - 3532313684 p^{3} T^{15} + 344384307 p^{4} T^{16} - 34516026 p^{5} T^{17} + 3426148 p^{6} T^{18} - 328911 p^{7} T^{19} + 29913 p^{8} T^{20} - 2276 p^{9} T^{21} + 186 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 21 T + 240 T^{2} + 2358 T^{3} + 24465 T^{4} + 273711 T^{5} + 2887160 T^{6} + 27063270 T^{7} + 262353663 T^{8} + 2738083932 T^{9} + 24030098235 T^{10} + 177764480820 T^{11} + 1406192625429 T^{12} + 177764480820 p T^{13} + 24030098235 p^{2} T^{14} + 2738083932 p^{3} T^{15} + 262353663 p^{4} T^{16} + 27063270 p^{5} T^{17} + 2887160 p^{6} T^{18} + 273711 p^{7} T^{19} + 24465 p^{8} T^{20} + 2358 p^{9} T^{21} + 240 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( 1 + 12 T - 21 T^{2} + 793 T^{3} + 23361 T^{4} + 78843 T^{5} + 735247 T^{6} + 33704739 T^{7} + 226888164 T^{8} + 713939650 T^{9} + 31480795200 T^{10} + 294372249243 T^{11} + 1100197019392 T^{12} + 294372249243 p T^{13} + 31480795200 p^{2} T^{14} + 713939650 p^{3} T^{15} + 226888164 p^{4} T^{16} + 33704739 p^{5} T^{17} + 735247 p^{6} T^{18} + 78843 p^{7} T^{19} + 23361 p^{8} T^{20} + 793 p^{9} T^{21} - 21 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.20792301084338657274745262102, −3.19998748740978983453311695798, −3.15516861636421851867962425119, −3.04648323457374325379816934716, −2.84412156533435121045439586665, −2.83833269408955384956497972314, −2.66932002829509761602312119378, −2.63851372297280956250664876446, −2.58419635992268454445859372086, −2.54498750295500218265423452041, −2.43792304623003983768801265251, −2.00881498712150820853862063334, −1.84116869113671053804583882412, −1.80829686228648276669424081989, −1.61657102127382554144289596392, −1.56570047824011246328203762106, −1.47757840641360789098729343560, −1.47071709179171851696291556603, −1.43393873286286976037114141155, −1.41152944153714230933293619812, −1.24268772299173432549447153068, −0.870426376688650794305197492475, −0.799544245176089800497542993092, −0.24513253960925232578859083948, −0.20384229118172381355196486345, 0.20384229118172381355196486345, 0.24513253960925232578859083948, 0.799544245176089800497542993092, 0.870426376688650794305197492475, 1.24268772299173432549447153068, 1.41152944153714230933293619812, 1.43393873286286976037114141155, 1.47071709179171851696291556603, 1.47757840641360789098729343560, 1.56570047824011246328203762106, 1.61657102127382554144289596392, 1.80829686228648276669424081989, 1.84116869113671053804583882412, 2.00881498712150820853862063334, 2.43792304623003983768801265251, 2.54498750295500218265423452041, 2.58419635992268454445859372086, 2.63851372297280956250664876446, 2.66932002829509761602312119378, 2.83833269408955384956497972314, 2.84412156533435121045439586665, 3.04648323457374325379816934716, 3.15516861636421851867962425119, 3.19998748740978983453311695798, 3.20792301084338657274745262102
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.