Dirichlet series
| L(s) = 1 | + 12·2-s + 12·3-s + 78·4-s + 5·5-s + 144·6-s + 6·7-s + 364·8-s + 78·9-s + 60·10-s + 7·11-s + 936·12-s + 9·13-s + 72·14-s + 60·15-s + 1.36e3·16-s + 25·17-s + 936·18-s − 12·19-s + 390·20-s + 72·21-s + 84·22-s + 22·23-s + 4.36e3·24-s − 25-s + 108·26-s + 364·27-s + 468·28-s + ⋯ |
| L(s) = 1 | + 8.48·2-s + 6.92·3-s + 39·4-s + 2.23·5-s + 58.7·6-s + 2.26·7-s + 128.·8-s + 26·9-s + 18.9·10-s + 2.11·11-s + 270.·12-s + 2.49·13-s + 19.2·14-s + 15.4·15-s + 341.·16-s + 6.06·17-s + 220.·18-s − 2.75·19-s + 87.2·20-s + 15.7·21-s + 17.9·22-s + 4.58·23-s + 891.·24-s − 1/5·25-s + 21.1·26-s + 70.0·27-s + 88.4·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 19^{12} \cdot 53^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 19^{12} \cdot 53^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{12} \cdot 19^{12} \cdot 53^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.59035\times 10^{20}\) |
| Root analytic conductor: | \(6.94590\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{12} \cdot 19^{12} \cdot 53^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(5.668516380\times10^{7}\) |
| \(L(\frac12)\) | \(\approx\) | \(5.668516380\times10^{7}\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{12} \) |
| 3 | \( ( 1 - T )^{12} \) | |
| 19 | \( ( 1 + T )^{12} \) | |
| 53 | \( ( 1 - T )^{12} \) | |
| good | 5 | \( 1 - p T + 26 T^{2} - 106 T^{3} + 403 T^{4} - 273 p T^{5} + 4314 T^{6} - 2561 p T^{7} + 35809 T^{8} - 94766 T^{9} + 47682 p T^{10} - 572569 T^{11} + 1307498 T^{12} - 572569 p T^{13} + 47682 p^{3} T^{14} - 94766 p^{3} T^{15} + 35809 p^{4} T^{16} - 2561 p^{6} T^{17} + 4314 p^{6} T^{18} - 273 p^{8} T^{19} + 403 p^{8} T^{20} - 106 p^{9} T^{21} + 26 p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 - 6 T + 6 p T^{2} - 26 p T^{3} + 842 T^{4} - 3126 T^{5} + 12234 T^{6} - 40841 T^{7} + 139737 T^{8} - 421349 T^{9} + 184260 p T^{10} - 505990 p T^{11} + 9887448 T^{12} - 505990 p^{2} T^{13} + 184260 p^{3} T^{14} - 421349 p^{3} T^{15} + 139737 p^{4} T^{16} - 40841 p^{5} T^{17} + 12234 p^{6} T^{18} - 3126 p^{7} T^{19} + 842 p^{8} T^{20} - 26 p^{10} T^{21} + 6 p^{11} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 7 T + 48 T^{2} - 194 T^{3} + 872 T^{4} - 3090 T^{5} + 1178 p T^{6} - 43921 T^{7} + 157899 T^{8} - 42890 p T^{9} + 1589506 T^{10} - 4941064 T^{11} + 17160504 T^{12} - 4941064 p T^{13} + 1589506 p^{2} T^{14} - 42890 p^{4} T^{15} + 157899 p^{4} T^{16} - 43921 p^{5} T^{17} + 1178 p^{7} T^{18} - 3090 p^{7} T^{19} + 872 p^{8} T^{20} - 194 p^{9} T^{21} + 48 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 9 T + 105 T^{2} - 654 T^{3} + 337 p T^{4} - 20957 T^{5} + 102874 T^{6} - 396562 T^{7} + 1578777 T^{8} - 5043901 T^{9} + 18041717 T^{10} - 52566721 T^{11} + 205552546 T^{12} - 52566721 p T^{13} + 18041717 p^{2} T^{14} - 5043901 p^{3} T^{15} + 1578777 p^{4} T^{16} - 396562 p^{5} T^{17} + 102874 p^{6} T^{18} - 20957 p^{7} T^{19} + 337 p^{9} T^{20} - 654 p^{9} T^{21} + 105 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 25 T + 401 T^{2} - 4764 T^{3} + 46653 T^{4} - 390107 T^{5} + 169080 p T^{6} - 18940060 T^{7} + 113180029 T^{8} - 617776027 T^{9} + 3101562359 T^{10} - 14365828137 T^{11} + 61574351498 T^{12} - 14365828137 p T^{13} + 3101562359 p^{2} T^{14} - 617776027 p^{3} T^{15} + 113180029 p^{4} T^{16} - 18940060 p^{5} T^{17} + 169080 p^{7} T^{18} - 390107 p^{7} T^{19} + 46653 p^{8} T^{20} - 4764 p^{9} T^{21} + 401 p^{10} T^{22} - 25 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 22 T + 359 T^{2} - 4313 T^{3} + 43687 T^{4} - 379849 T^{5} + 2940135 T^{6} - 20552284 T^{7} + 131883227 T^{8} - 783898094 T^{9} + 4359914946 T^{10} - 990285552 p T^{11} + 112444041402 T^{12} - 990285552 p^{2} T^{13} + 4359914946 p^{2} T^{14} - 783898094 p^{3} T^{15} + 131883227 p^{4} T^{16} - 20552284 p^{5} T^{17} + 2940135 p^{6} T^{18} - 379849 p^{7} T^{19} + 43687 p^{8} T^{20} - 4313 p^{9} T^{21} + 359 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - T + 94 T^{2} - 211 T^{3} + 7101 T^{4} - 10302 T^{5} + 378037 T^{6} - 607867 T^{7} + 16144533 T^{8} - 19970146 T^{9} + 601798241 T^{10} - 699227997 T^{11} + 18215670146 T^{12} - 699227997 p T^{13} + 601798241 p^{2} T^{14} - 19970146 p^{3} T^{15} + 16144533 p^{4} T^{16} - 607867 p^{5} T^{17} + 378037 p^{6} T^{18} - 10302 p^{7} T^{19} + 7101 p^{8} T^{20} - 211 p^{9} T^{21} + 94 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 23 T + 445 T^{2} - 5917 T^{3} + 70703 T^{4} - 698985 T^{5} + 6397901 T^{6} - 1662893 p T^{7} + 391319819 T^{8} - 2693480316 T^{9} + 17667357270 T^{10} - 106680145964 T^{11} + 617566877258 T^{12} - 106680145964 p T^{13} + 17667357270 p^{2} T^{14} - 2693480316 p^{3} T^{15} + 391319819 p^{4} T^{16} - 1662893 p^{6} T^{17} + 6397901 p^{6} T^{18} - 698985 p^{7} T^{19} + 70703 p^{8} T^{20} - 5917 p^{9} T^{21} + 445 p^{10} T^{22} - 23 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + T + 153 T^{2} + 172 T^{3} + 14557 T^{4} + 17741 T^{5} + 1027620 T^{6} + 1357490 T^{7} + 57856685 T^{8} + 79884153 T^{9} + 2726402803 T^{10} + 3714046391 T^{11} + 109011214090 T^{12} + 3714046391 p T^{13} + 2726402803 p^{2} T^{14} + 79884153 p^{3} T^{15} + 57856685 p^{4} T^{16} + 1357490 p^{5} T^{17} + 1027620 p^{6} T^{18} + 17741 p^{7} T^{19} + 14557 p^{8} T^{20} + 172 p^{9} T^{21} + 153 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 15 T + 388 T^{2} + 4794 T^{3} + 72092 T^{4} + 745880 T^{5} + 8424369 T^{6} + 74587795 T^{7} + 691255809 T^{8} + 5325124642 T^{9} + 42167491899 T^{10} + 285108637746 T^{11} + 1967933018180 T^{12} + 285108637746 p T^{13} + 42167491899 p^{2} T^{14} + 5325124642 p^{3} T^{15} + 691255809 p^{4} T^{16} + 74587795 p^{5} T^{17} + 8424369 p^{6} T^{18} + 745880 p^{7} T^{19} + 72092 p^{8} T^{20} + 4794 p^{9} T^{21} + 388 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 2 T + 233 T^{2} - 287 T^{3} + 25751 T^{4} - 2581 T^{5} + 1822165 T^{6} + 2603366 T^{7} + 94853259 T^{8} + 308963208 T^{9} + 4122552186 T^{10} + 19951457638 T^{11} + 173311159930 T^{12} + 19951457638 p T^{13} + 4122552186 p^{2} T^{14} + 308963208 p^{3} T^{15} + 94853259 p^{4} T^{16} + 2603366 p^{5} T^{17} + 1822165 p^{6} T^{18} - 2581 p^{7} T^{19} + 25751 p^{8} T^{20} - 287 p^{9} T^{21} + 233 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 11 T + 8 p T^{2} - 3620 T^{3} + 66852 T^{4} - 558038 T^{5} + 7433127 T^{6} - 1152645 p T^{7} + 586439367 T^{8} - 3799206258 T^{9} + 35897346273 T^{10} - 211671258188 T^{11} + 1825396231912 T^{12} - 211671258188 p T^{13} + 35897346273 p^{2} T^{14} - 3799206258 p^{3} T^{15} + 586439367 p^{4} T^{16} - 1152645 p^{6} T^{17} + 7433127 p^{6} T^{18} - 558038 p^{7} T^{19} + 66852 p^{8} T^{20} - 3620 p^{9} T^{21} + 8 p^{11} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 3 T + 248 T^{2} - 369 T^{3} + 39031 T^{4} - 19410 T^{5} + 4366205 T^{6} + 1216027 T^{7} + 397020031 T^{8} + 288715740 T^{9} + 29711899243 T^{10} + 27402516177 T^{11} + 1904841582242 T^{12} + 27402516177 p T^{13} + 29711899243 p^{2} T^{14} + 288715740 p^{3} T^{15} + 397020031 p^{4} T^{16} + 1216027 p^{5} T^{17} + 4366205 p^{6} T^{18} - 19410 p^{7} T^{19} + 39031 p^{8} T^{20} - 369 p^{9} T^{21} + 248 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 16 T + 524 T^{2} - 6836 T^{3} + 129571 T^{4} - 1439034 T^{5} + 20509529 T^{6} - 199095192 T^{7} + 2345272305 T^{8} - 20188893860 T^{9} + 205080277363 T^{10} - 1573519403630 T^{11} + 14070845097590 T^{12} - 1573519403630 p T^{13} + 205080277363 p^{2} T^{14} - 20188893860 p^{3} T^{15} + 2345272305 p^{4} T^{16} - 199095192 p^{5} T^{17} + 20509529 p^{6} T^{18} - 1439034 p^{7} T^{19} + 129571 p^{8} T^{20} - 6836 p^{9} T^{21} + 524 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 2 T + 441 T^{2} + 807 T^{3} + 100313 T^{4} + 185435 T^{5} + 15467773 T^{6} + 28350280 T^{7} + 1794818515 T^{8} + 3152450438 T^{9} + 164806364754 T^{10} + 268293161168 T^{11} + 12238599670182 T^{12} + 268293161168 p T^{13} + 164806364754 p^{2} T^{14} + 3152450438 p^{3} T^{15} + 1794818515 p^{4} T^{16} + 28350280 p^{5} T^{17} + 15467773 p^{6} T^{18} + 185435 p^{7} T^{19} + 100313 p^{8} T^{20} + 807 p^{9} T^{21} + 441 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 4 T + 295 T^{2} + 1171 T^{3} + 48916 T^{4} + 157817 T^{5} + 5913703 T^{6} + 16676588 T^{7} + 596288815 T^{8} + 1528431904 T^{9} + 52773216194 T^{10} + 126752953158 T^{11} + 4029652229688 T^{12} + 126752953158 p T^{13} + 52773216194 p^{2} T^{14} + 1528431904 p^{3} T^{15} + 596288815 p^{4} T^{16} + 16676588 p^{5} T^{17} + 5913703 p^{6} T^{18} + 157817 p^{7} T^{19} + 48916 p^{8} T^{20} + 1171 p^{9} T^{21} + 295 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 35 T + 14 p T^{2} - 19802 T^{3} + 337228 T^{4} - 4649414 T^{5} + 59031846 T^{6} - 663910779 T^{7} + 7186192935 T^{8} - 72635374546 T^{9} + 715614460732 T^{10} - 6614507243496 T^{11} + 58598570501080 T^{12} - 6614507243496 p T^{13} + 715614460732 p^{2} T^{14} - 72635374546 p^{3} T^{15} + 7186192935 p^{4} T^{16} - 663910779 p^{5} T^{17} + 59031846 p^{6} T^{18} - 4649414 p^{7} T^{19} + 337228 p^{8} T^{20} - 19802 p^{9} T^{21} + 14 p^{11} T^{22} - 35 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 4 T + 426 T^{2} - 1191 T^{3} + 93776 T^{4} - 169540 T^{5} + 13766285 T^{6} - 9364565 T^{7} + 1510680577 T^{8} + 618883155 T^{9} + 136819141585 T^{10} + 159025960167 T^{11} + 11137472530060 T^{12} + 159025960167 p T^{13} + 136819141585 p^{2} T^{14} + 618883155 p^{3} T^{15} + 1510680577 p^{4} T^{16} - 9364565 p^{5} T^{17} + 13766285 p^{6} T^{18} - 169540 p^{7} T^{19} + 93776 p^{8} T^{20} - 1191 p^{9} T^{21} + 426 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 39 T + 1409 T^{2} - 34372 T^{3} + 751607 T^{4} - 13605761 T^{5} + 223273646 T^{6} - 3231604216 T^{7} + 42940334829 T^{8} - 516698001793 T^{9} + 5759209830497 T^{10} - 58811067007505 T^{11} + 558833153717942 T^{12} - 58811067007505 p T^{13} + 5759209830497 p^{2} T^{14} - 516698001793 p^{3} T^{15} + 42940334829 p^{4} T^{16} - 3231604216 p^{5} T^{17} + 223273646 p^{6} T^{18} - 13605761 p^{7} T^{19} + 751607 p^{8} T^{20} - 34372 p^{9} T^{21} + 1409 p^{10} T^{22} - 39 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 11 T + 813 T^{2} - 8213 T^{3} + 318501 T^{4} - 2929739 T^{5} + 79424657 T^{6} - 661176333 T^{7} + 14041054675 T^{8} - 105095625576 T^{9} + 1854779740530 T^{10} - 12370218718928 T^{11} + 187815033681166 T^{12} - 12370218718928 p T^{13} + 1854779740530 p^{2} T^{14} - 105095625576 p^{3} T^{15} + 14041054675 p^{4} T^{16} - 661176333 p^{5} T^{17} + 79424657 p^{6} T^{18} - 2929739 p^{7} T^{19} + 318501 p^{8} T^{20} - 8213 p^{9} T^{21} + 813 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 21 T + 608 T^{2} + 11944 T^{3} + 214638 T^{4} + 3522096 T^{5} + 51788336 T^{6} + 716763737 T^{7} + 9197836255 T^{8} + 110748578026 T^{9} + 1260503811952 T^{10} + 13463903393776 T^{11} + 136885072090724 T^{12} + 13463903393776 p T^{13} + 1260503811952 p^{2} T^{14} + 110748578026 p^{3} T^{15} + 9197836255 p^{4} T^{16} + 716763737 p^{5} T^{17} + 51788336 p^{6} T^{18} + 3522096 p^{7} T^{19} + 214638 p^{8} T^{20} + 11944 p^{9} T^{21} + 608 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.64514062640254816672599724901, −2.29550180797706036386950579915, −2.26808368006076189776690349321, −2.22420331350330854661767754844, −2.21681045007349175255951743564, −2.03914971134598557868817380806, −1.99073969422985338808499050277, −1.97895564238242722481611287350, −1.92744650197104116043156202521, −1.89544200992479898322622494902, −1.88703687734192841277064287689, −1.85140957571321333843445193060, −1.77107881204988379028351169743, −1.48605309156844843959808620997, −1.38228651929654063212554528654, −1.32887236382943866038069092007, −1.20161474345917723160803058552, −1.13810722007725777910731618674, −1.11504197651370158066496049723, −1.06644112837339972644064715416, −1.04453227490818807476811603182, −1.00286417959859929690669785131, −0.848916519713600525283971448273, −0.68310278312737429819363521006, −0.61819255987411233985208614741, 0.61819255987411233985208614741, 0.68310278312737429819363521006, 0.848916519713600525283971448273, 1.00286417959859929690669785131, 1.04453227490818807476811603182, 1.06644112837339972644064715416, 1.11504197651370158066496049723, 1.13810722007725777910731618674, 1.20161474345917723160803058552, 1.32887236382943866038069092007, 1.38228651929654063212554528654, 1.48605309156844843959808620997, 1.77107881204988379028351169743, 1.85140957571321333843445193060, 1.88703687734192841277064287689, 1.89544200992479898322622494902, 1.92744650197104116043156202521, 1.97895564238242722481611287350, 1.99073969422985338808499050277, 2.03914971134598557868817380806, 2.21681045007349175255951743564, 2.22420331350330854661767754844, 2.26808368006076189776690349321, 2.29550180797706036386950579915, 2.64514062640254816672599724901
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.