Dirichlet series
| L(s) = 1 | + 6·2-s + 15·4-s + 3·5-s − 6·7-s + 18·8-s + 18·10-s − 3·11-s − 6·13-s − 36·14-s + 3·16-s − 9·17-s − 3·19-s + 45·20-s − 18·22-s + 12·23-s + 6·25-s − 36·26-s − 90·28-s + 6·29-s + 3·31-s − 30·32-s − 54·34-s − 18·35-s − 3·37-s − 18·38-s + 54·40-s − 15·41-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 15/2·4-s + 1.34·5-s − 2.26·7-s + 6.36·8-s + 5.69·10-s − 0.904·11-s − 1.66·13-s − 9.62·14-s + 3/4·16-s − 2.18·17-s − 0.688·19-s + 10.0·20-s − 3.83·22-s + 2.50·23-s + 6/5·25-s − 7.06·26-s − 17.0·28-s + 1.11·29-s + 0.538·31-s − 5.30·32-s − 9.26·34-s − 3.04·35-s − 0.493·37-s − 2.91·38-s + 8.53·40-s − 2.34·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{48}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.00535976\) |
| Root analytic conductor: | \(0.804231\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{48} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.483248288\) |
| \(L(\frac12)\) | \(\approx\) | \(2.483248288\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 - 3 T + 3 T^{2} - 3 T^{4} + 3 p T^{5} - 11 T^{6} + 3 p^{2} T^{7} - 3 p^{2} T^{8} + 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )( 1 - 3 T + 9 T^{2} - 9 p T^{3} + 9 p^{2} T^{4} - 57 T^{5} + 91 T^{6} - 57 p T^{7} + 9 p^{4} T^{8} - 9 p^{4} T^{9} + 9 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} ) \) |
| 5 | \( 1 - 3 T + 3 T^{2} - 18 T^{3} + 87 T^{4} - 147 T^{5} + 323 T^{6} - 1368 T^{7} + 3096 T^{8} - 5562 T^{9} + 16272 T^{10} - 8262 p T^{11} + 82629 T^{12} - 8262 p^{2} T^{13} + 16272 p^{2} T^{14} - 5562 p^{3} T^{15} + 3096 p^{4} T^{16} - 1368 p^{5} T^{17} + 323 p^{6} T^{18} - 147 p^{7} T^{19} + 87 p^{8} T^{20} - 18 p^{9} T^{21} + 3 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 + 6 T + 12 T^{2} - 11 T^{3} - 213 T^{4} - 678 T^{5} - 32 p T^{6} + 3942 T^{7} + 15255 T^{8} + 25135 T^{9} - 22044 T^{10} - 210732 T^{11} - 647141 T^{12} - 210732 p T^{13} - 22044 p^{2} T^{14} + 25135 p^{3} T^{15} + 15255 p^{4} T^{16} + 3942 p^{5} T^{17} - 32 p^{7} T^{18} - 678 p^{7} T^{19} - 213 p^{8} T^{20} - 11 p^{9} T^{21} + 12 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 3 T - 15 T^{2} - 126 T^{3} - 201 T^{4} + 1488 T^{5} + 7145 T^{6} + 1530 T^{7} - 5634 p T^{8} - 202716 T^{9} - 19692 T^{10} + 1304451 T^{11} + 4526883 T^{12} + 1304451 p T^{13} - 19692 p^{2} T^{14} - 202716 p^{3} T^{15} - 5634 p^{5} T^{16} + 1530 p^{5} T^{17} + 7145 p^{6} T^{18} + 1488 p^{7} T^{19} - 201 p^{8} T^{20} - 126 p^{9} T^{21} - 15 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 6 T + 48 T^{2} + 214 T^{3} + 1488 T^{4} + 456 p T^{5} + 32329 T^{6} + 112023 T^{7} + 560277 T^{8} + 1799710 T^{9} + 8467593 T^{10} + 25055985 T^{11} + 112181629 T^{12} + 25055985 p T^{13} + 8467593 p^{2} T^{14} + 1799710 p^{3} T^{15} + 560277 p^{4} T^{16} + 112023 p^{5} T^{17} + 32329 p^{6} T^{18} + 456 p^{8} T^{19} + 1488 p^{8} T^{20} + 214 p^{9} T^{21} + 48 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 9 T - 30 T^{2} - 423 T^{3} + 1029 T^{4} + 14184 T^{5} - 23521 T^{6} - 296649 T^{7} + 637560 T^{8} + 4620213 T^{9} - 12537675 T^{10} - 28264410 T^{11} + 250681641 T^{12} - 28264410 p T^{13} - 12537675 p^{2} T^{14} + 4620213 p^{3} T^{15} + 637560 p^{4} T^{16} - 296649 p^{5} T^{17} - 23521 p^{6} T^{18} + 14184 p^{7} T^{19} + 1029 p^{8} T^{20} - 423 p^{9} T^{21} - 30 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 3 T - 75 T^{2} - 242 T^{3} + 3012 T^{4} + 9714 T^{5} - 85589 T^{6} - 257166 T^{7} + 1946502 T^{8} + 4391737 T^{9} - 39399504 T^{10} - 1762662 p T^{11} + 40166287 p T^{12} - 1762662 p^{2} T^{13} - 39399504 p^{2} T^{14} + 4391737 p^{3} T^{15} + 1946502 p^{4} T^{16} - 257166 p^{5} T^{17} - 85589 p^{6} T^{18} + 9714 p^{7} T^{19} + 3012 p^{8} T^{20} - 242 p^{9} T^{21} - 75 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 12 T + 48 T^{2} - 153 T^{3} - 336 T^{4} + 12228 T^{5} - 51922 T^{6} + 116820 T^{7} - 165330 T^{8} - 4324509 T^{9} + 14509764 T^{10} + 2453454 T^{11} + 107316369 T^{12} + 2453454 p T^{13} + 14509764 p^{2} T^{14} - 4324509 p^{3} T^{15} - 165330 p^{4} T^{16} + 116820 p^{5} T^{17} - 51922 p^{6} T^{18} + 12228 p^{7} T^{19} - 336 p^{8} T^{20} - 153 p^{9} T^{21} + 48 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 6 T + 21 T^{2} - 252 T^{3} + 249 T^{4} + 984 T^{5} + 18431 T^{6} - 29592 T^{7} + 680634 T^{8} - 5882274 T^{9} + 10161684 T^{10} - 17557326 T^{11} + 254066229 T^{12} - 17557326 p T^{13} + 10161684 p^{2} T^{14} - 5882274 p^{3} T^{15} + 680634 p^{4} T^{16} - 29592 p^{5} T^{17} + 18431 p^{6} T^{18} + 984 p^{7} T^{19} + 249 p^{8} T^{20} - 252 p^{9} T^{21} + 21 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 3 T + 84 T^{2} - 14 p T^{3} + 5601 T^{4} - 30963 T^{5} + 266473 T^{6} - 1627992 T^{7} + 11453211 T^{8} - 69240287 T^{9} + 408317577 T^{10} - 2527882269 T^{11} + 13547586181 T^{12} - 2527882269 p T^{13} + 408317577 p^{2} T^{14} - 69240287 p^{3} T^{15} + 11453211 p^{4} T^{16} - 1627992 p^{5} T^{17} + 266473 p^{6} T^{18} - 30963 p^{7} T^{19} + 5601 p^{8} T^{20} - 14 p^{10} T^{21} + 84 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 3 T - 156 T^{2} - 107 T^{3} + 13731 T^{4} - 9132 T^{5} - 864755 T^{6} + 641043 T^{7} + 43249536 T^{8} - 18536771 T^{9} - 1953626739 T^{10} + 269355786 T^{11} + 78884071369 T^{12} + 269355786 p T^{13} - 1953626739 p^{2} T^{14} - 18536771 p^{3} T^{15} + 43249536 p^{4} T^{16} + 641043 p^{5} T^{17} - 864755 p^{6} T^{18} - 9132 p^{7} T^{19} + 13731 p^{8} T^{20} - 107 p^{9} T^{21} - 156 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 15 T + 93 T^{2} + 90 T^{3} - 60 p T^{4} - 12513 T^{5} + 27971 T^{6} + 441396 T^{7} + 3206862 T^{8} + 12736494 T^{9} - 41813613 T^{10} - 1731506832 T^{11} - 16389887967 T^{12} - 1731506832 p T^{13} - 41813613 p^{2} T^{14} + 12736494 p^{3} T^{15} + 3206862 p^{4} T^{16} + 441396 p^{5} T^{17} + 27971 p^{6} T^{18} - 12513 p^{7} T^{19} - 60 p^{9} T^{20} + 90 p^{9} T^{21} + 93 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 3 T - 60 T^{2} + 16 T^{3} + 606 T^{4} + 5874 T^{5} + 2983 p T^{6} - 73818 T^{7} - 5307417 T^{8} - 22910987 T^{9} - 79924800 T^{10} + 580797228 T^{11} + 14492410483 T^{12} + 580797228 p T^{13} - 79924800 p^{2} T^{14} - 22910987 p^{3} T^{15} - 5307417 p^{4} T^{16} - 73818 p^{5} T^{17} + 2983 p^{7} T^{18} + 5874 p^{7} T^{19} + 606 p^{8} T^{20} + 16 p^{9} T^{21} - 60 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 15 T + 111 T^{2} - 873 T^{3} + 6828 T^{4} - 69612 T^{5} + 654227 T^{6} - 4732173 T^{7} + 31522707 T^{8} - 170376048 T^{9} + 1258782219 T^{10} - 11485670769 T^{11} + 82734051465 T^{12} - 11485670769 p T^{13} + 1258782219 p^{2} T^{14} - 170376048 p^{3} T^{15} + 31522707 p^{4} T^{16} - 4732173 p^{5} T^{17} + 654227 p^{6} T^{18} - 69612 p^{7} T^{19} + 6828 p^{8} T^{20} - 873 p^{9} T^{21} + 111 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( ( 1 - 9 T + 210 T^{2} - 1872 T^{3} + 23856 T^{4} - 168327 T^{5} + 1634317 T^{6} - 168327 p T^{7} + 23856 p^{2} T^{8} - 1872 p^{3} T^{9} + 210 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 - 12 T + 192 T^{2} - 2349 T^{3} + 25089 T^{4} - 223824 T^{5} + 1972808 T^{6} - 12709350 T^{7} + 71501877 T^{8} - 339681357 T^{9} - 107943444 T^{10} + 13247965206 T^{11} - 122980417173 T^{12} + 13247965206 p T^{13} - 107943444 p^{2} T^{14} - 339681357 p^{3} T^{15} + 71501877 p^{4} T^{16} - 12709350 p^{5} T^{17} + 1972808 p^{6} T^{18} - 223824 p^{7} T^{19} + 25089 p^{8} T^{20} - 2349 p^{9} T^{21} + 192 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 12 T - 51 T^{2} + 583 T^{3} + 2127 T^{4} + 45474 T^{5} - 455363 T^{6} - 2399139 T^{7} + 11507670 T^{8} + 53383966 T^{9} + 844033821 T^{10} - 2578276122 T^{11} - 56950876769 T^{12} - 2578276122 p T^{13} + 844033821 p^{2} T^{14} + 53383966 p^{3} T^{15} + 11507670 p^{4} T^{16} - 2399139 p^{5} T^{17} - 455363 p^{6} T^{18} + 45474 p^{7} T^{19} + 2127 p^{8} T^{20} + 583 p^{9} T^{21} - 51 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 15 T + 255 T^{2} + 2968 T^{3} + 36174 T^{4} + 397221 T^{5} + 4107115 T^{6} + 41367024 T^{7} + 386429292 T^{8} + 3556146616 T^{9} + 31289775603 T^{10} + 264760435272 T^{11} + 2216964278029 T^{12} + 264760435272 p T^{13} + 31289775603 p^{2} T^{14} + 3556146616 p^{3} T^{15} + 386429292 p^{4} T^{16} + 41367024 p^{5} T^{17} + 4107115 p^{6} T^{18} + 397221 p^{7} T^{19} + 36174 p^{8} T^{20} + 2968 p^{9} T^{21} + 255 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 27 T + 78 T^{2} - 2565 T^{3} + 13071 T^{4} + 524664 T^{5} - 751711 T^{6} - 30297321 T^{7} + 410765508 T^{8} + 3391054713 T^{9} - 30034133541 T^{10} - 13624108308 T^{11} + 3600759258249 T^{12} - 13624108308 p T^{13} - 30034133541 p^{2} T^{14} + 3391054713 p^{3} T^{15} + 410765508 p^{4} T^{16} - 30297321 p^{5} T^{17} - 751711 p^{6} T^{18} + 524664 p^{7} T^{19} + 13071 p^{8} T^{20} - 2565 p^{9} T^{21} + 78 p^{10} T^{22} + 27 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 6 T - 228 T^{2} + 2296 T^{3} + 24945 T^{4} - 381255 T^{5} - 980072 T^{6} + 40200363 T^{7} - 102286134 T^{8} - 2648934335 T^{9} + 21743689350 T^{10} + 78452536893 T^{11} - 2017821540323 T^{12} + 78452536893 p T^{13} + 21743689350 p^{2} T^{14} - 2648934335 p^{3} T^{15} - 102286134 p^{4} T^{16} + 40200363 p^{5} T^{17} - 980072 p^{6} T^{18} - 381255 p^{7} T^{19} + 24945 p^{8} T^{20} + 2296 p^{9} T^{21} - 228 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 42 T + 813 T^{2} + 9520 T^{3} + 72840 T^{4} + 356811 T^{5} + 973207 T^{6} - 893781 T^{7} - 62793603 T^{8} - 1355379536 T^{9} - 23955645108 T^{10} - 329298299862 T^{11} - 3388931313773 T^{12} - 329298299862 p T^{13} - 23955645108 p^{2} T^{14} - 1355379536 p^{3} T^{15} - 62793603 p^{4} T^{16} - 893781 p^{5} T^{17} + 973207 p^{6} T^{18} + 356811 p^{7} T^{19} + 72840 p^{8} T^{20} + 9520 p^{9} T^{21} + 813 p^{10} T^{22} + 42 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 39 T + 912 T^{2} + 16200 T^{3} + 251079 T^{4} + 3515997 T^{5} + 45358019 T^{6} + 541131408 T^{7} + 6100532325 T^{8} + 65514800025 T^{9} + 671478204717 T^{10} + 6541571603403 T^{11} + 60933732837525 T^{12} + 6541571603403 p T^{13} + 671478204717 p^{2} T^{14} + 65514800025 p^{3} T^{15} + 6100532325 p^{4} T^{16} + 541131408 p^{5} T^{17} + 45358019 p^{6} T^{18} + 3515997 p^{7} T^{19} + 251079 p^{8} T^{20} + 16200 p^{9} T^{21} + 912 p^{10} T^{22} + 39 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 9 T - 273 T^{2} - 2772 T^{3} + 38802 T^{4} + 449316 T^{5} - 3561871 T^{6} - 54551502 T^{7} + 157767516 T^{8} + 4371660207 T^{9} + 3816883044 T^{10} - 152630961444 T^{11} - 900621732009 T^{12} - 152630961444 p T^{13} + 3816883044 p^{2} T^{14} + 4371660207 p^{3} T^{15} + 157767516 p^{4} T^{16} - 54551502 p^{5} T^{17} - 3561871 p^{6} T^{18} + 449316 p^{7} T^{19} + 38802 p^{8} T^{20} - 2772 p^{9} T^{21} - 273 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 3 T + 102 T^{2} - 1010 T^{3} + 156 p T^{4} - 13512 T^{5} + 1127323 T^{6} - 7762230 T^{7} + 240327 p T^{8} + 575063737 T^{9} + 1596748254 T^{10} + 54554445012 T^{11} - 1453572795209 T^{12} + 54554445012 p T^{13} + 1596748254 p^{2} T^{14} + 575063737 p^{3} T^{15} + 240327 p^{5} T^{16} - 7762230 p^{5} T^{17} + 1127323 p^{6} T^{18} - 13512 p^{7} T^{19} + 156 p^{9} T^{20} - 1010 p^{9} T^{21} + 102 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−5.29266228537374159111820897485, −5.23343862859717802680960239048, −4.95526482619938213764591098792, −4.89889003732022284234093291981, −4.85924815146617231226069224434, −4.83137412678742614477004440869, −4.69640221119488074749108962563, −4.50418679033153046507861486613, −4.46187122974390436386343906454, −4.09237583792113210719002491654, −4.02206801099561134492346145659, −3.93545098214888768680473724978, −3.80003426340643955048360263003, −3.79087644557024191915964516651, −3.69122795591946895028701248533, −3.16761777815462144155723855216, −3.05982508720670982471144215024, −2.89633575768505488749041643482, −2.78558948457901225601709292024, −2.73169922799128222426771254845, −2.66954639704832544177155038989, −2.37098045680480032471313580067, −2.16068297297437154876442698701, −1.86540728154967913402049276079, −1.36238949171048255313363429519, 1.36238949171048255313363429519, 1.86540728154967913402049276079, 2.16068297297437154876442698701, 2.37098045680480032471313580067, 2.66954639704832544177155038989, 2.73169922799128222426771254845, 2.78558948457901225601709292024, 2.89633575768505488749041643482, 3.05982508720670982471144215024, 3.16761777815462144155723855216, 3.69122795591946895028701248533, 3.79087644557024191915964516651, 3.80003426340643955048360263003, 3.93545098214888768680473724978, 4.02206801099561134492346145659, 4.09237583792113210719002491654, 4.46187122974390436386343906454, 4.50418679033153046507861486613, 4.69640221119488074749108962563, 4.83137412678742614477004440869, 4.85924815146617231226069224434, 4.89889003732022284234093291981, 4.95526482619938213764591098792, 5.23343862859717802680960239048, 5.29266228537374159111820897485
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.