Dirichlet series
| L(s) = 1 | − 4·3-s + 3·9-s + 11-s − 4·13-s − 8·17-s − 9·19-s + 27-s + 8·29-s − 33·31-s − 4·33-s − 6·37-s + 16·39-s + 27·41-s − 50·43-s + 18·47-s − 36·49-s + 32·51-s − 22·53-s + 36·57-s − 33·59-s − 8·61-s − 41·67-s − 19·71-s + 5·73-s − 58·79-s + 14·81-s − 18·83-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 9-s + 0.301·11-s − 1.10·13-s − 1.94·17-s − 2.06·19-s + 0.192·27-s + 1.48·29-s − 5.92·31-s − 0.696·33-s − 0.986·37-s + 2.56·39-s + 4.21·41-s − 7.62·43-s + 2.62·47-s − 5.14·49-s + 4.48·51-s − 3.02·53-s + 4.76·57-s − 4.29·59-s − 1.02·61-s − 5.00·67-s − 2.25·71-s + 0.585·73-s − 6.52·79-s + 14/9·81-s − 1.97·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{48}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 5^{48}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.71931\times 10^{22}\) |
| Root analytic conductor: | \(8.93590\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 2^{48} \cdot 5^{48} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
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 | 2 | \( 1 \) |
| 5 | \( 1 \) | |
| good | 3 | \( 1 + 4 T + 13 T^{2} + 13 p T^{3} + 11 p^{2} T^{4} + 242 T^{5} + 199 p T^{6} + 1297 T^{7} + 899 p T^{8} + 1831 p T^{9} + 10363 T^{10} + 19181 T^{11} + 34687 T^{12} + 19181 p T^{13} + 10363 p^{2} T^{14} + 1831 p^{4} T^{15} + 899 p^{5} T^{16} + 1297 p^{5} T^{17} + 199 p^{7} T^{18} + 242 p^{7} T^{19} + 11 p^{10} T^{20} + 13 p^{10} T^{21} + 13 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 + 36 T^{2} + 29 T^{3} + 698 T^{4} + 908 T^{5} + 9868 T^{6} + 15478 T^{7} + 109118 T^{8} + 186454 T^{9} + 990516 T^{10} + 1676023 T^{11} + 7566134 T^{12} + 1676023 p T^{13} + 990516 p^{2} T^{14} + 186454 p^{3} T^{15} + 109118 p^{4} T^{16} + 15478 p^{5} T^{17} + 9868 p^{6} T^{18} + 908 p^{7} T^{19} + 698 p^{8} T^{20} + 29 p^{9} T^{21} + 36 p^{10} T^{22} + p^{12} T^{24} \) | |
| 11 | \( 1 - T + 64 T^{2} - 73 T^{3} + 2039 T^{4} - 3225 T^{5} + 43442 T^{6} - 94748 T^{7} + 699360 T^{8} - 1936042 T^{9} + 9262534 T^{10} - 28737116 T^{11} + 107530031 T^{12} - 28737116 p T^{13} + 9262534 p^{2} T^{14} - 1936042 p^{3} T^{15} + 699360 p^{4} T^{16} - 94748 p^{5} T^{17} + 43442 p^{6} T^{18} - 3225 p^{7} T^{19} + 2039 p^{8} T^{20} - 73 p^{9} T^{21} + 64 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 4 T + 67 T^{2} + 258 T^{3} + 2371 T^{4} + 651 p T^{5} + 55544 T^{6} + 186748 T^{7} + 968596 T^{8} + 3126820 T^{9} + 13915321 T^{10} + 44396727 T^{11} + 183438872 T^{12} + 44396727 p T^{13} + 13915321 p^{2} T^{14} + 3126820 p^{3} T^{15} + 968596 p^{4} T^{16} + 186748 p^{5} T^{17} + 55544 p^{6} T^{18} + 651 p^{8} T^{19} + 2371 p^{8} T^{20} + 258 p^{9} T^{21} + 67 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 8 T + 98 T^{2} + 803 T^{3} + 350 p T^{4} + 41016 T^{5} + 248671 T^{6} + 1456068 T^{7} + 7714817 T^{8} + 38970524 T^{9} + 185755162 T^{10} + 825755519 T^{11} + 3531244737 T^{12} + 825755519 p T^{13} + 185755162 p^{2} T^{14} + 38970524 p^{3} T^{15} + 7714817 p^{4} T^{16} + 1456068 p^{5} T^{17} + 248671 p^{6} T^{18} + 41016 p^{7} T^{19} + 350 p^{9} T^{20} + 803 p^{9} T^{21} + 98 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 9 T + 155 T^{2} + 1059 T^{3} + 10439 T^{4} + 59361 T^{5} + 437519 T^{6} + 2217885 T^{7} + 13594224 T^{8} + 63825234 T^{9} + 341518040 T^{10} + 1487971815 T^{11} + 7117203145 T^{12} + 1487971815 p T^{13} + 341518040 p^{2} T^{14} + 63825234 p^{3} T^{15} + 13594224 p^{4} T^{16} + 2217885 p^{5} T^{17} + 437519 p^{6} T^{18} + 59361 p^{7} T^{19} + 10439 p^{8} T^{20} + 1059 p^{9} T^{21} + 155 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 133 T^{2} + 41 T^{3} + 9583 T^{4} + 6391 T^{5} + 480955 T^{6} + 422490 T^{7} + 18548142 T^{8} + 18043432 T^{9} + 574999252 T^{10} + 551954396 T^{11} + 14573638316 T^{12} + 551954396 p T^{13} + 574999252 p^{2} T^{14} + 18043432 p^{3} T^{15} + 18548142 p^{4} T^{16} + 422490 p^{5} T^{17} + 480955 p^{6} T^{18} + 6391 p^{7} T^{19} + 9583 p^{8} T^{20} + 41 p^{9} T^{21} + 133 p^{10} T^{22} + p^{12} T^{24} \) | |
| 29 | \( 1 - 8 T + 231 T^{2} - 1631 T^{3} + 24954 T^{4} - 157335 T^{5} + 1688878 T^{6} - 9643786 T^{7} + 81700890 T^{8} - 429110156 T^{9} + 3085657251 T^{10} - 15062618512 T^{11} + 96914311686 T^{12} - 15062618512 p T^{13} + 3085657251 p^{2} T^{14} - 429110156 p^{3} T^{15} + 81700890 p^{4} T^{16} - 9643786 p^{5} T^{17} + 1688878 p^{6} T^{18} - 157335 p^{7} T^{19} + 24954 p^{8} T^{20} - 1631 p^{9} T^{21} + 231 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 33 T + 691 T^{2} + 10590 T^{3} + 133177 T^{4} + 1425378 T^{5} + 13478198 T^{6} + 114436707 T^{7} + 887830438 T^{8} + 6341142258 T^{9} + 42049628083 T^{10} + 259484352348 T^{11} + 1495541083832 T^{12} + 259484352348 p T^{13} + 42049628083 p^{2} T^{14} + 6341142258 p^{3} T^{15} + 887830438 p^{4} T^{16} + 114436707 p^{5} T^{17} + 13478198 p^{6} T^{18} + 1425378 p^{7} T^{19} + 133177 p^{8} T^{20} + 10590 p^{9} T^{21} + 691 p^{10} T^{22} + 33 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 6 T + 161 T^{2} + 699 T^{3} + 13127 T^{4} + 36003 T^{5} + 683886 T^{6} + 529233 T^{7} + 25077310 T^{8} - 52597581 T^{9} + 704022109 T^{10} - 4320130416 T^{11} + 21361608460 T^{12} - 4320130416 p T^{13} + 704022109 p^{2} T^{14} - 52597581 p^{3} T^{15} + 25077310 p^{4} T^{16} + 529233 p^{5} T^{17} + 683886 p^{6} T^{18} + 36003 p^{7} T^{19} + 13127 p^{8} T^{20} + 699 p^{9} T^{21} + 161 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 27 T + 511 T^{2} - 7009 T^{3} + 84998 T^{4} - 897036 T^{5} + 8800935 T^{6} - 78175149 T^{7} + 654220084 T^{8} - 5068327010 T^{9} + 37445653862 T^{10} - 258847952683 T^{11} + 1711735793201 T^{12} - 258847952683 p T^{13} + 37445653862 p^{2} T^{14} - 5068327010 p^{3} T^{15} + 654220084 p^{4} T^{16} - 78175149 p^{5} T^{17} + 8800935 p^{6} T^{18} - 897036 p^{7} T^{19} + 84998 p^{8} T^{20} - 7009 p^{9} T^{21} + 511 p^{10} T^{22} - 27 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 50 T + 1514 T^{2} + 32952 T^{3} + 572028 T^{4} + 8270181 T^{5} + 103012532 T^{6} + 1127556284 T^{7} + 11030590343 T^{8} + 97526527163 T^{9} + 786835084254 T^{10} + 5826874485744 T^{11} + 39778675109609 T^{12} + 5826874485744 p T^{13} + 786835084254 p^{2} T^{14} + 97526527163 p^{3} T^{15} + 11030590343 p^{4} T^{16} + 1127556284 p^{5} T^{17} + 103012532 p^{6} T^{18} + 8270181 p^{7} T^{19} + 572028 p^{8} T^{20} + 32952 p^{9} T^{21} + 1514 p^{10} T^{22} + 50 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 18 T + 410 T^{2} - 5901 T^{3} + 83744 T^{4} - 971058 T^{5} + 10796709 T^{6} - 105447435 T^{7} + 983244118 T^{8} - 8330341887 T^{9} + 67344470785 T^{10} - 502103746155 T^{11} + 3577491385570 T^{12} - 502103746155 p T^{13} + 67344470785 p^{2} T^{14} - 8330341887 p^{3} T^{15} + 983244118 p^{4} T^{16} - 105447435 p^{5} T^{17} + 10796709 p^{6} T^{18} - 971058 p^{7} T^{19} + 83744 p^{8} T^{20} - 5901 p^{9} T^{21} + 410 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 22 T + 457 T^{2} + 6721 T^{3} + 90944 T^{4} + 1044897 T^{5} + 11393369 T^{6} + 111413249 T^{7} + 1046142912 T^{8} + 9047324083 T^{9} + 75436785802 T^{10} + 586109042444 T^{11} + 4414396994294 T^{12} + 586109042444 p T^{13} + 75436785802 p^{2} T^{14} + 9047324083 p^{3} T^{15} + 1046142912 p^{4} T^{16} + 111413249 p^{5} T^{17} + 11393369 p^{6} T^{18} + 1044897 p^{7} T^{19} + 90944 p^{8} T^{20} + 6721 p^{9} T^{21} + 457 p^{10} T^{22} + 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 33 T + 907 T^{2} + 16735 T^{3} + 271738 T^{4} + 3553595 T^{5} + 42204748 T^{6} + 429556308 T^{7} + 4082302191 T^{8} + 34595563571 T^{9} + 284993681062 T^{10} + 2190341581024 T^{11} + 17171216492687 T^{12} + 2190341581024 p T^{13} + 284993681062 p^{2} T^{14} + 34595563571 p^{3} T^{15} + 4082302191 p^{4} T^{16} + 429556308 p^{5} T^{17} + 42204748 p^{6} T^{18} + 3553595 p^{7} T^{19} + 271738 p^{8} T^{20} + 16735 p^{9} T^{21} + 907 p^{10} T^{22} + 33 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 8 T + 511 T^{2} + 3730 T^{3} + 127597 T^{4} + 848793 T^{5} + 20628538 T^{6} + 124660062 T^{7} + 2404682358 T^{8} + 13137590578 T^{9} + 212838723343 T^{10} + 1042997890913 T^{11} + 14662002738312 T^{12} + 1042997890913 p T^{13} + 212838723343 p^{2} T^{14} + 13137590578 p^{3} T^{15} + 2404682358 p^{4} T^{16} + 124660062 p^{5} T^{17} + 20628538 p^{6} T^{18} + 848793 p^{7} T^{19} + 127597 p^{8} T^{20} + 3730 p^{9} T^{21} + 511 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 41 T + 985 T^{2} + 17261 T^{3} + 251726 T^{4} + 3224686 T^{5} + 37670243 T^{6} + 408248089 T^{7} + 4174854470 T^{8} + 40540376726 T^{9} + 375606442960 T^{10} + 3312530963585 T^{11} + 27819070848299 T^{12} + 3312530963585 p T^{13} + 375606442960 p^{2} T^{14} + 40540376726 p^{3} T^{15} + 4174854470 p^{4} T^{16} + 408248089 p^{5} T^{17} + 37670243 p^{6} T^{18} + 3224686 p^{7} T^{19} + 251726 p^{8} T^{20} + 17261 p^{9} T^{21} + 985 p^{10} T^{22} + 41 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 19 T + 649 T^{2} + 9281 T^{3} + 192316 T^{4} + 2294426 T^{5} + 36385502 T^{6} + 376126849 T^{7} + 4944261688 T^{8} + 45030115336 T^{9} + 509535232201 T^{10} + 4114412755115 T^{11} + 40856991146054 T^{12} + 4114412755115 p T^{13} + 509535232201 p^{2} T^{14} + 45030115336 p^{3} T^{15} + 4944261688 p^{4} T^{16} + 376126849 p^{5} T^{17} + 36385502 p^{6} T^{18} + 2294426 p^{7} T^{19} + 192316 p^{8} T^{20} + 9281 p^{9} T^{21} + 649 p^{10} T^{22} + 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 5 T + 458 T^{2} - 2986 T^{3} + 112078 T^{4} - 789701 T^{5} + 19012225 T^{6} - 133739750 T^{7} + 2430332457 T^{8} - 16439815297 T^{9} + 244814864957 T^{10} - 1539571700756 T^{11} + 19864201439391 T^{12} - 1539571700756 p T^{13} + 244814864957 p^{2} T^{14} - 16439815297 p^{3} T^{15} + 2430332457 p^{4} T^{16} - 133739750 p^{5} T^{17} + 19012225 p^{6} T^{18} - 789701 p^{7} T^{19} + 112078 p^{8} T^{20} - 2986 p^{9} T^{21} + 458 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 58 T + 2011 T^{2} + 50168 T^{3} + 1008451 T^{4} + 17088777 T^{5} + 253671718 T^{6} + 3365253222 T^{7} + 40687556388 T^{8} + 453247328372 T^{9} + 4705322084479 T^{10} + 45756819215545 T^{11} + 419089671848064 T^{12} + 45756819215545 p T^{13} + 4705322084479 p^{2} T^{14} + 453247328372 p^{3} T^{15} + 40687556388 p^{4} T^{16} + 3365253222 p^{5} T^{17} + 253671718 p^{6} T^{18} + 17088777 p^{7} T^{19} + 1008451 p^{8} T^{20} + 50168 p^{9} T^{21} + 2011 p^{10} T^{22} + 58 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 18 T + 755 T^{2} + 11652 T^{3} + 271796 T^{4} + 3674293 T^{5} + 62108527 T^{6} + 742897578 T^{7} + 10075932185 T^{8} + 107138640433 T^{9} + 1227416927515 T^{10} + 11595639812875 T^{11} + 115442423898019 T^{12} + 11595639812875 p T^{13} + 1227416927515 p^{2} T^{14} + 107138640433 p^{3} T^{15} + 10075932185 p^{4} T^{16} + 742897578 p^{5} T^{17} + 62108527 p^{6} T^{18} + 3674293 p^{7} T^{19} + 271796 p^{8} T^{20} + 11652 p^{9} T^{21} + 755 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 44 T + 1400 T^{2} - 30884 T^{3} + 569149 T^{4} - 8633736 T^{5} + 116046754 T^{6} - 1369780830 T^{7} + 14849705789 T^{8} - 147312899579 T^{9} + 1398353007025 T^{10} - 12853778568515 T^{11} + 120415772141655 T^{12} - 12853778568515 p T^{13} + 1398353007025 p^{2} T^{14} - 147312899579 p^{3} T^{15} + 14849705789 p^{4} T^{16} - 1369780830 p^{5} T^{17} + 116046754 p^{6} T^{18} - 8633736 p^{7} T^{19} + 569149 p^{8} T^{20} - 30884 p^{9} T^{21} + 1400 p^{10} T^{22} - 44 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 22 T + 990 T^{2} + 17139 T^{3} + 445704 T^{4} + 6444867 T^{5} + 124463599 T^{6} + 1550152900 T^{7} + 24338658558 T^{8} + 265033336823 T^{9} + 3530994630690 T^{10} + 33795267261715 T^{11} + 390618666138445 T^{12} + 33795267261715 p T^{13} + 3530994630690 p^{2} T^{14} + 265033336823 p^{3} T^{15} + 24338658558 p^{4} T^{16} + 1550152900 p^{5} T^{17} + 124463599 p^{6} T^{18} + 6444867 p^{7} T^{19} + 445704 p^{8} T^{20} + 17139 p^{9} T^{21} + 990 p^{10} T^{22} + 22 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.80856660263215822854951649930, −2.52688860297331088436659706885, −2.37890617523688140896807369323, −2.28512463071911249382866420603, −2.26858657516990238333587523118, −2.22808898937280711624430553913, −2.17986523984512285298385217885, −2.15689198590434660013920112583, −2.09803526204314906008671648476, −1.91931303874710532698478238552, −1.80670350599827110130485417023, −1.74776990112132255111268308737, −1.73489238375750046925294680798, −1.73463744994730343371409585791, −1.65815343954149641166252266030, −1.61202509834739310831260552922, −1.36251584957084959594039619554, −1.25124209363863678627550725929, −1.24521850239786994941663417458, −1.19137358184628197853000974460, −1.14804446735543912384388682093, −1.12178895845873542870869167081, −1.00529192067118392458841473067, −0.920326662956658491741932507986, −0.70236922958958895142353373907, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.70236922958958895142353373907, 0.920326662956658491741932507986, 1.00529192067118392458841473067, 1.12178895845873542870869167081, 1.14804446735543912384388682093, 1.19137358184628197853000974460, 1.24521850239786994941663417458, 1.25124209363863678627550725929, 1.36251584957084959594039619554, 1.61202509834739310831260552922, 1.65815343954149641166252266030, 1.73463744994730343371409585791, 1.73489238375750046925294680798, 1.74776990112132255111268308737, 1.80670350599827110130485417023, 1.91931303874710532698478238552, 2.09803526204314906008671648476, 2.15689198590434660013920112583, 2.17986523984512285298385217885, 2.22808898937280711624430553913, 2.26858657516990238333587523118, 2.28512463071911249382866420603, 2.37890617523688140896807369323, 2.52688860297331088436659706885, 2.80856660263215822854951649930
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.