Dirichlet series
| L(s) = 1 | + 12·2-s + 78·4-s + 3·5-s + 6·7-s + 364·8-s + 36·10-s + 10·11-s + 5·13-s + 72·14-s + 1.36e3·16-s + 8·17-s + 2·19-s + 234·20-s + 120·22-s + 9·23-s − 22·25-s + 60·26-s + 468·28-s + 19·29-s + 10·31-s + 4.36e3·32-s + 96·34-s + 18·35-s + 11·37-s + 24·38-s + 1.09e3·40-s + 8·41-s + ⋯ |
| L(s) = 1 | + 8.48·2-s + 39·4-s + 1.34·5-s + 2.26·7-s + 128.·8-s + 11.3·10-s + 3.01·11-s + 1.38·13-s + 19.2·14-s + 341.·16-s + 1.94·17-s + 0.458·19-s + 52.3·20-s + 25.5·22-s + 1.87·23-s − 4.39·25-s + 11.7·26-s + 88.4·28-s + 3.52·29-s + 1.79·31-s + 772.·32-s + 16.4·34-s + 3.04·35-s + 1.80·37-s + 3.89·38-s + 172.·40-s + 1.24·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36} \cdot 149^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36} \cdot 149^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{36} \cdot 149^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.94635\times 10^{21}\) |
| Root analytic conductor: | \(8.01546\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{36} \cdot 149^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.230110016\times10^{6}\) |
| \(L(\frac12)\) | \(\approx\) | \(2.230110016\times10^{6}\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| 149 | \( ( 1 - T )^{12} \) | |
| good | 5 | \( 1 - 3 T + 31 T^{2} - 89 T^{3} + 104 p T^{4} - 1429 T^{5} + 6032 T^{6} - 622 p^{2} T^{7} + 53006 T^{8} - 126152 T^{9} + 367406 T^{10} - 795897 T^{11} + 2044683 T^{12} - 795897 p T^{13} + 367406 p^{2} T^{14} - 126152 p^{3} T^{15} + 53006 p^{4} T^{16} - 622 p^{7} T^{17} + 6032 p^{6} T^{18} - 1429 p^{7} T^{19} + 104 p^{9} T^{20} - 89 p^{9} T^{21} + 31 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 - 6 T + 59 T^{2} - 36 p T^{3} + 1506 T^{4} - 5272 T^{5} + 3519 p T^{6} - 75456 T^{7} + 299403 T^{8} - 16787 p^{2} T^{9} + 2869172 T^{10} - 7125598 T^{11} + 22245057 T^{12} - 7125598 p T^{13} + 2869172 p^{2} T^{14} - 16787 p^{5} T^{15} + 299403 p^{4} T^{16} - 75456 p^{5} T^{17} + 3519 p^{7} T^{18} - 5272 p^{7} T^{19} + 1506 p^{8} T^{20} - 36 p^{10} T^{21} + 59 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 10 T + 119 T^{2} - 852 T^{3} + 6179 T^{4} - 34957 T^{5} + 194642 T^{6} - 921276 T^{7} + 4266253 T^{8} - 17433663 T^{9} + 69581584 T^{10} - 249077190 T^{11} + 870352803 T^{12} - 249077190 p T^{13} + 69581584 p^{2} T^{14} - 17433663 p^{3} T^{15} + 4266253 p^{4} T^{16} - 921276 p^{5} T^{17} + 194642 p^{6} T^{18} - 34957 p^{7} T^{19} + 6179 p^{8} T^{20} - 852 p^{9} T^{21} + 119 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 5 T + 83 T^{2} - 389 T^{3} + 3299 T^{4} - 14252 T^{5} + 84327 T^{6} - 25390 p T^{7} + 1590748 T^{8} - 5592589 T^{9} + 24482767 T^{10} - 78779092 T^{11} + 332956381 T^{12} - 78779092 p T^{13} + 24482767 p^{2} T^{14} - 5592589 p^{3} T^{15} + 1590748 p^{4} T^{16} - 25390 p^{6} T^{17} + 84327 p^{6} T^{18} - 14252 p^{7} T^{19} + 3299 p^{8} T^{20} - 389 p^{9} T^{21} + 83 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 8 T + 163 T^{2} - 1083 T^{3} + 12073 T^{4} - 68054 T^{5} + 547343 T^{6} - 2664751 T^{7} + 17308162 T^{8} - 256176 p^{2} T^{9} + 413119042 T^{10} - 1576525489 T^{11} + 7818022337 T^{12} - 1576525489 p T^{13} + 413119042 p^{2} T^{14} - 256176 p^{5} T^{15} + 17308162 p^{4} T^{16} - 2664751 p^{5} T^{17} + 547343 p^{6} T^{18} - 68054 p^{7} T^{19} + 12073 p^{8} T^{20} - 1083 p^{9} T^{21} + 163 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 2 T + 143 T^{2} - 226 T^{3} + 10027 T^{4} - 12501 T^{5} + 459424 T^{6} - 451004 T^{7} + 15440053 T^{8} - 12114979 T^{9} + 404393020 T^{10} - 265891932 T^{11} + 8522163709 T^{12} - 265891932 p T^{13} + 404393020 p^{2} T^{14} - 12114979 p^{3} T^{15} + 15440053 p^{4} T^{16} - 451004 p^{5} T^{17} + 459424 p^{6} T^{18} - 12501 p^{7} T^{19} + 10027 p^{8} T^{20} - 226 p^{9} T^{21} + 143 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 9 T + 164 T^{2} - 1157 T^{3} + 13061 T^{4} - 78224 T^{5} + 689511 T^{6} - 3640497 T^{7} + 27275253 T^{8} - 129726161 T^{9} + 856762847 T^{10} - 3689448901 T^{11} + 21824414143 T^{12} - 3689448901 p T^{13} + 856762847 p^{2} T^{14} - 129726161 p^{3} T^{15} + 27275253 p^{4} T^{16} - 3640497 p^{5} T^{17} + 689511 p^{6} T^{18} - 78224 p^{7} T^{19} + 13061 p^{8} T^{20} - 1157 p^{9} T^{21} + 164 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 19 T + 421 T^{2} - 5467 T^{3} + 72294 T^{4} - 725015 T^{5} + 7169556 T^{6} - 58779582 T^{7} + 470430092 T^{8} - 3247459554 T^{9} + 21808912434 T^{10} - 128624114567 T^{11} + 737186448151 T^{12} - 128624114567 p T^{13} + 21808912434 p^{2} T^{14} - 3247459554 p^{3} T^{15} + 470430092 p^{4} T^{16} - 58779582 p^{5} T^{17} + 7169556 p^{6} T^{18} - 725015 p^{7} T^{19} + 72294 p^{8} T^{20} - 5467 p^{9} T^{21} + 421 p^{10} T^{22} - 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 10 T + 238 T^{2} - 2081 T^{3} + 28109 T^{4} - 210477 T^{5} + 2139213 T^{6} - 13941262 T^{7} + 117503039 T^{8} - 680726339 T^{9} + 4999690949 T^{10} - 26064900855 T^{11} + 171532558799 T^{12} - 26064900855 p T^{13} + 4999690949 p^{2} T^{14} - 680726339 p^{3} T^{15} + 117503039 p^{4} T^{16} - 13941262 p^{5} T^{17} + 2139213 p^{6} T^{18} - 210477 p^{7} T^{19} + 28109 p^{8} T^{20} - 2081 p^{9} T^{21} + 238 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 11 T + 359 T^{2} - 3519 T^{3} + 62339 T^{4} - 540523 T^{5} + 6839425 T^{6} - 52371342 T^{7} + 525133134 T^{8} - 3545120461 T^{9} + 29642085735 T^{10} - 175713509903 T^{11} + 1259572750809 T^{12} - 175713509903 p T^{13} + 29642085735 p^{2} T^{14} - 3545120461 p^{3} T^{15} + 525133134 p^{4} T^{16} - 52371342 p^{5} T^{17} + 6839425 p^{6} T^{18} - 540523 p^{7} T^{19} + 62339 p^{8} T^{20} - 3519 p^{9} T^{21} + 359 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 8 T + 280 T^{2} - 1912 T^{3} + 37858 T^{4} - 239703 T^{5} + 3422732 T^{6} - 20732444 T^{7} + 233518280 T^{8} - 1353253845 T^{9} + 12753687572 T^{10} - 69492949936 T^{11} + 573947757762 T^{12} - 69492949936 p T^{13} + 12753687572 p^{2} T^{14} - 1353253845 p^{3} T^{15} + 233518280 p^{4} T^{16} - 20732444 p^{5} T^{17} + 3422732 p^{6} T^{18} - 239703 p^{7} T^{19} + 37858 p^{8} T^{20} - 1912 p^{9} T^{21} + 280 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 13 T + 314 T^{2} - 3115 T^{3} + 43775 T^{4} - 356232 T^{5} + 3818346 T^{6} - 27341201 T^{7} + 251135363 T^{8} - 1668841681 T^{9} + 13812272101 T^{10} - 85899802546 T^{11} + 646688616071 T^{12} - 85899802546 p T^{13} + 13812272101 p^{2} T^{14} - 1668841681 p^{3} T^{15} + 251135363 p^{4} T^{16} - 27341201 p^{5} T^{17} + 3818346 p^{6} T^{18} - 356232 p^{7} T^{19} + 43775 p^{8} T^{20} - 3115 p^{9} T^{21} + 314 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 11 T + 380 T^{2} - 3348 T^{3} + 67142 T^{4} - 501867 T^{5} + 7617229 T^{6} - 49927967 T^{7} + 632089918 T^{8} - 3697798992 T^{9} + 40883587123 T^{10} - 215405763898 T^{11} + 2129597328809 T^{12} - 215405763898 p T^{13} + 40883587123 p^{2} T^{14} - 3697798992 p^{3} T^{15} + 632089918 p^{4} T^{16} - 49927967 p^{5} T^{17} + 7617229 p^{6} T^{18} - 501867 p^{7} T^{19} + 67142 p^{8} T^{20} - 3348 p^{9} T^{21} + 380 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 24 T + 603 T^{2} - 9679 T^{3} + 149521 T^{4} - 1854987 T^{5} + 21952133 T^{6} - 225761694 T^{7} + 2221765972 T^{8} - 19721645009 T^{9} + 168377353725 T^{10} - 1321000801602 T^{11} + 10002935068151 T^{12} - 1321000801602 p T^{13} + 168377353725 p^{2} T^{14} - 19721645009 p^{3} T^{15} + 2221765972 p^{4} T^{16} - 225761694 p^{5} T^{17} + 21952133 p^{6} T^{18} - 1854987 p^{7} T^{19} + 149521 p^{8} T^{20} - 9679 p^{9} T^{21} + 603 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 10 T + 471 T^{2} - 4552 T^{3} + 110446 T^{4} - 1016167 T^{5} + 16993766 T^{6} - 146376854 T^{7} + 1901091247 T^{8} - 15078221015 T^{9} + 162209617518 T^{10} - 1164196223873 T^{11} + 10798200421241 T^{12} - 1164196223873 p T^{13} + 162209617518 p^{2} T^{14} - 15078221015 p^{3} T^{15} + 1901091247 p^{4} T^{16} - 146376854 p^{5} T^{17} + 16993766 p^{6} T^{18} - 1016167 p^{7} T^{19} + 110446 p^{8} T^{20} - 4552 p^{9} T^{21} + 471 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 518 T^{2} - 300 T^{3} + 126924 T^{4} - 139383 T^{5} + 19680217 T^{6} - 29906974 T^{7} + 2183730146 T^{8} - 3899695138 T^{9} + 185807889183 T^{10} - 341309649871 T^{11} + 12608534769183 T^{12} - 341309649871 p T^{13} + 185807889183 p^{2} T^{14} - 3899695138 p^{3} T^{15} + 2183730146 p^{4} T^{16} - 29906974 p^{5} T^{17} + 19680217 p^{6} T^{18} - 139383 p^{7} T^{19} + 126924 p^{8} T^{20} - 300 p^{9} T^{21} + 518 p^{10} T^{22} + p^{12} T^{24} \) | |
| 67 | \( 1 - 21 T + 799 T^{2} - 12751 T^{3} + 276774 T^{4} - 3570999 T^{5} + 56863559 T^{6} - 615634366 T^{7} + 7907297962 T^{8} - 73592160483 T^{9} + 799798334931 T^{10} - 6491628578230 T^{11} + 61208138531903 T^{12} - 6491628578230 p T^{13} + 799798334931 p^{2} T^{14} - 73592160483 p^{3} T^{15} + 7907297962 p^{4} T^{16} - 615634366 p^{5} T^{17} + 56863559 p^{6} T^{18} - 3570999 p^{7} T^{19} + 276774 p^{8} T^{20} - 12751 p^{9} T^{21} + 799 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 37 T + 1111 T^{2} - 22875 T^{3} + 415447 T^{4} - 6245078 T^{5} + 86287478 T^{6} - 1055184800 T^{7} + 12132103572 T^{8} - 127257281273 T^{9} + 1267733176411 T^{10} - 11660132106472 T^{11} + 102111645443079 T^{12} - 11660132106472 p T^{13} + 1267733176411 p^{2} T^{14} - 127257281273 p^{3} T^{15} + 12132103572 p^{4} T^{16} - 1055184800 p^{5} T^{17} + 86287478 p^{6} T^{18} - 6245078 p^{7} T^{19} + 415447 p^{8} T^{20} - 22875 p^{9} T^{21} + 1111 p^{10} T^{22} - 37 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 2 T + 502 T^{2} + 1073 T^{3} + 125653 T^{4} + 313232 T^{5} + 20818320 T^{6} + 60283783 T^{7} + 2565761663 T^{8} + 8188988785 T^{9} + 250651940189 T^{10} + 809922763303 T^{11} + 20087838486865 T^{12} + 809922763303 p T^{13} + 250651940189 p^{2} T^{14} + 8188988785 p^{3} T^{15} + 2565761663 p^{4} T^{16} + 60283783 p^{5} T^{17} + 20818320 p^{6} T^{18} + 313232 p^{7} T^{19} + 125653 p^{8} T^{20} + 1073 p^{9} T^{21} + 502 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 7 T + 700 T^{2} - 3679 T^{3} + 228154 T^{4} - 879730 T^{5} + 46628737 T^{6} - 127937112 T^{7} + 6779437914 T^{8} - 13119162148 T^{9} + 752285146813 T^{10} - 1100360658298 T^{11} + 66267884019181 T^{12} - 1100360658298 p T^{13} + 752285146813 p^{2} T^{14} - 13119162148 p^{3} T^{15} + 6779437914 p^{4} T^{16} - 127937112 p^{5} T^{17} + 46628737 p^{6} T^{18} - 879730 p^{7} T^{19} + 228154 p^{8} T^{20} - 3679 p^{9} T^{21} + 700 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 22 T + 719 T^{2} - 10556 T^{3} + 199296 T^{4} - 2156997 T^{5} + 30929344 T^{6} - 264149042 T^{7} + 3374891639 T^{8} - 24768853573 T^{9} + 314751114974 T^{10} - 2148093090825 T^{11} + 27214698406967 T^{12} - 2148093090825 p T^{13} + 314751114974 p^{2} T^{14} - 24768853573 p^{3} T^{15} + 3374891639 p^{4} T^{16} - 264149042 p^{5} T^{17} + 30929344 p^{6} T^{18} - 2156997 p^{7} T^{19} + 199296 p^{8} T^{20} - 10556 p^{9} T^{21} + 719 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 40 T + 1430 T^{2} - 35157 T^{3} + 772387 T^{4} - 14151415 T^{5} + 235920178 T^{6} - 3483529268 T^{7} + 47407646992 T^{8} - 586085327676 T^{9} + 6728255791535 T^{10} - 70958208182375 T^{11} + 697284512185649 T^{12} - 70958208182375 p T^{13} + 6728255791535 p^{2} T^{14} - 586085327676 p^{3} T^{15} + 47407646992 p^{4} T^{16} - 3483529268 p^{5} T^{17} + 235920178 p^{6} T^{18} - 14151415 p^{7} T^{19} + 772387 p^{8} T^{20} - 35157 p^{9} T^{21} + 1430 p^{10} T^{22} - 40 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 7 T + 517 T^{2} - 4705 T^{3} + 141436 T^{4} - 1468706 T^{5} + 27804818 T^{6} - 301005861 T^{7} + 4295214504 T^{8} - 46235717659 T^{9} + 541114581696 T^{10} - 5592020727862 T^{11} + 57075181715471 T^{12} - 5592020727862 p T^{13} + 541114581696 p^{2} T^{14} - 46235717659 p^{3} T^{15} + 4295214504 p^{4} T^{16} - 301005861 p^{5} T^{17} + 27804818 p^{6} T^{18} - 1468706 p^{7} T^{19} + 141436 p^{8} T^{20} - 4705 p^{9} T^{21} + 517 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.43660903677284458173983256641, −2.21431609407764790918467735526, −2.14521265789877067684077579955, −2.02650193183669209468743227458, −2.00081549017571213793728034811, −1.96625480050564142542837112356, −1.95366963943081658081755542159, −1.86393666065914296036118195635, −1.82343920267248112587229260349, −1.81478004097303145159321636138, −1.74302256335895659800330226066, −1.71606224615788730879969725654, −1.63960047149821116377783565681, −1.31922313803195601836768428607, −1.14791720882009693228513994613, −1.12919971501791974702763863626, −1.04843236294126568789944709147, −1.00039901242079406580438160079, −0.940644388369713509827820596410, −0.870621774805142521334446803767, −0.810302465441579329971638412352, −0.808604698520744401459235028687, −0.62590415231152585427114275687, −0.49128028165741792510735569521, −0.48809188807862084564068635480, 0.48809188807862084564068635480, 0.49128028165741792510735569521, 0.62590415231152585427114275687, 0.808604698520744401459235028687, 0.810302465441579329971638412352, 0.870621774805142521334446803767, 0.940644388369713509827820596410, 1.00039901242079406580438160079, 1.04843236294126568789944709147, 1.12919971501791974702763863626, 1.14791720882009693228513994613, 1.31922313803195601836768428607, 1.63960047149821116377783565681, 1.71606224615788730879969725654, 1.74302256335895659800330226066, 1.81478004097303145159321636138, 1.82343920267248112587229260349, 1.86393666065914296036118195635, 1.95366963943081658081755542159, 1.96625480050564142542837112356, 2.00081549017571213793728034811, 2.02650193183669209468743227458, 2.14521265789877067684077579955, 2.21431609407764790918467735526, 2.43660903677284458173983256641
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.