Dirichlet series
| L(s) = 1 | + 6·3-s − 3·5-s + 6·7-s + 18·9-s − 3·11-s − 6·13-s − 18·15-s + 9·17-s + 3·19-s + 36·21-s + 12·23-s + 6·25-s + 39·27-s − 6·29-s − 3·31-s − 18·33-s − 18·35-s − 3·37-s − 36·39-s + 15·41-s − 3·43-s − 54·45-s + 15·47-s + 24·49-s + 54·51-s − 18·53-s + 9·55-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 1.34·5-s + 2.26·7-s + 6·9-s − 0.904·11-s − 1.66·13-s − 4.64·15-s + 2.18·17-s + 0.688·19-s + 7.85·21-s + 2.50·23-s + 6/5·25-s + 7.50·27-s − 1.11·29-s − 0.538·31-s − 3.13·33-s − 3.04·35-s − 0.493·37-s − 5.76·39-s + 2.34·41-s − 0.457·43-s − 8.04·45-s + 2.18·47-s + 24/7·49-s + 7.56·51-s − 2.47·53-s + 1.21·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 3^{36}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.83877\times 10^{6}\) |
| Root analytic conductor: | \(1.85729\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 3^{36} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(11.93305079\) |
| \(L(\frac12)\) | \(\approx\) | \(11.93305079\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 2 p T + 2 p^{2} T^{2} - 13 p T^{3} + 7 p^{2} T^{4} - p^{4} T^{5} + 13 p^{2} T^{6} - p^{5} T^{7} + 7 p^{4} T^{8} - 13 p^{4} T^{9} + 2 p^{6} T^{10} - 2 p^{6} T^{11} + p^{6} T^{12} \) | |
| good | 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
−3.67714917361753093060934232014, −3.64530158848651561155314154203, −3.56948438096814540745352514423, −3.48585001159102816200903703958, −3.18687972796385590946623696280, −3.09926662392709452389310410958, −3.06886955640931298277789236553, −2.99934454109806908446165053933, −2.80628336165834024115115930579, −2.73154452989583862208565013108, −2.55182322370590435894195993184, −2.52823719641886417167109628495, −2.47671162099533558754071985177, −2.39963629794728371896681502478, −2.20279525681424345109437818477, −2.08105118570981725687988071911, −2.05009275352463876703087103378, −1.62414877588048692117009808849, −1.55225038835295249907134847453, −1.35491781042669014145476312652, −1.30592209775080635955812140695, −1.06976758893814698218807919890, −1.00243087488254085998436137029, −0.833339566874523011935366150290, −0.22001651112520579866512229005, 0.22001651112520579866512229005, 0.833339566874523011935366150290, 1.00243087488254085998436137029, 1.06976758893814698218807919890, 1.30592209775080635955812140695, 1.35491781042669014145476312652, 1.55225038835295249907134847453, 1.62414877588048692117009808849, 2.05009275352463876703087103378, 2.08105118570981725687988071911, 2.20279525681424345109437818477, 2.39963629794728371896681502478, 2.47671162099533558754071985177, 2.52823719641886417167109628495, 2.55182322370590435894195993184, 2.73154452989583862208565013108, 2.80628336165834024115115930579, 2.99934454109806908446165053933, 3.06886955640931298277789236553, 3.09926662392709452389310410958, 3.18687972796385590946623696280, 3.48585001159102816200903703958, 3.56948438096814540745352514423, 3.64530158848651561155314154203, 3.67714917361753093060934232014
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.