Dirichlet series
| L(s) = 1 | − 3·2-s − 5·3-s − 5·4-s − 7·5-s + 15·6-s − 20·7-s + 25·8-s − 4·9-s + 21·10-s − 11·11-s + 25·12-s − 11·13-s + 60·14-s + 35·15-s − 2·16-s + 17-s + 12·18-s − 34·19-s + 35·20-s + 100·21-s + 33·22-s − 7·23-s − 125·24-s − 2·25-s + 33·26-s + 66·27-s + 100·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 2.88·3-s − 5/2·4-s − 3.13·5-s + 6.12·6-s − 7.55·7-s + 8.83·8-s − 4/3·9-s + 6.64·10-s − 3.31·11-s + 7.21·12-s − 3.05·13-s + 16.0·14-s + 9.03·15-s − 1/2·16-s + 0.242·17-s + 2.82·18-s − 7.80·19-s + 7.82·20-s + 21.8·21-s + 7.03·22-s − 1.45·23-s − 25.5·24-s − 2/5·25-s + 6.47·26-s + 12.7·27-s + 18.8·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(401^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(401^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(401^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.16160\times 10^{6}\) |
| Root analytic conductor: | \(1.78941\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 401^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
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 | 401 | \( ( 1 + T )^{12} \) |
| good | 2 | \( 1 + 3 T + 7 p T^{2} + p^{5} T^{3} + 93 T^{4} + 177 T^{5} + 25 p^{4} T^{6} + 667 T^{7} + 1281 T^{8} + 241 p^{3} T^{9} + 3299 T^{10} + 2283 p T^{11} + 893 p^{3} T^{12} + 2283 p^{2} T^{13} + 3299 p^{2} T^{14} + 241 p^{6} T^{15} + 1281 p^{4} T^{16} + 667 p^{5} T^{17} + 25 p^{10} T^{18} + 177 p^{7} T^{19} + 93 p^{8} T^{20} + p^{14} T^{21} + 7 p^{11} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) |
| 3 | \( 1 + 5 T + 29 T^{2} + 11 p^{2} T^{3} + 13 p^{3} T^{4} + 314 p T^{5} + 287 p^{2} T^{6} + 1954 p T^{7} + 4532 p T^{8} + 3023 p^{2} T^{9} + 55720 T^{10} + 3715 p^{3} T^{11} + 184832 T^{12} + 3715 p^{4} T^{13} + 55720 p^{2} T^{14} + 3023 p^{5} T^{15} + 4532 p^{5} T^{16} + 1954 p^{6} T^{17} + 287 p^{8} T^{18} + 314 p^{8} T^{19} + 13 p^{11} T^{20} + 11 p^{11} T^{21} + 29 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 5 | \( 1 + 7 T + 51 T^{2} + 243 T^{3} + 1107 T^{4} + 823 p T^{5} + 14524 T^{6} + 9004 p T^{7} + 26673 p T^{8} + 358917 T^{9} + 930437 T^{10} + 2229078 T^{11} + 5170791 T^{12} + 2229078 p T^{13} + 930437 p^{2} T^{14} + 358917 p^{3} T^{15} + 26673 p^{5} T^{16} + 9004 p^{6} T^{17} + 14524 p^{6} T^{18} + 823 p^{8} T^{19} + 1107 p^{8} T^{20} + 243 p^{9} T^{21} + 51 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 + 20 T + 239 T^{2} + 2056 T^{3} + 14130 T^{4} + 81236 T^{5} + 404273 T^{6} + 1776956 T^{7} + 7009656 T^{8} + 3580624 p T^{9} + 81871798 T^{10} + 35049568 p T^{11} + 676539007 T^{12} + 35049568 p^{2} T^{13} + 81871798 p^{2} T^{14} + 3580624 p^{4} T^{15} + 7009656 p^{4} T^{16} + 1776956 p^{5} T^{17} + 404273 p^{6} T^{18} + 81236 p^{7} T^{19} + 14130 p^{8} T^{20} + 2056 p^{9} T^{21} + 239 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + p T + 123 T^{2} + 80 p T^{3} + 6167 T^{4} + 3139 p T^{5} + 188779 T^{6} + 886552 T^{7} + 4064410 T^{8} + 16519517 T^{9} + 65587493 T^{10} + 234235173 T^{11} + 817402503 T^{12} + 234235173 p T^{13} + 65587493 p^{2} T^{14} + 16519517 p^{3} T^{15} + 4064410 p^{4} T^{16} + 886552 p^{5} T^{17} + 188779 p^{6} T^{18} + 3139 p^{8} T^{19} + 6167 p^{8} T^{20} + 80 p^{10} T^{21} + 123 p^{10} T^{22} + p^{12} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 11 T + 139 T^{2} + 1039 T^{3} + 7894 T^{4} + 45128 T^{5} + 257863 T^{6} + 1202832 T^{7} + 5659044 T^{8} + 22662011 T^{9} + 93689294 T^{10} + 339292283 T^{11} + 1298859290 T^{12} + 339292283 p T^{13} + 93689294 p^{2} T^{14} + 22662011 p^{3} T^{15} + 5659044 p^{4} T^{16} + 1202832 p^{5} T^{17} + 257863 p^{6} T^{18} + 45128 p^{7} T^{19} + 7894 p^{8} T^{20} + 1039 p^{9} T^{21} + 139 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - T + 83 T^{2} - 189 T^{3} + 3652 T^{4} - 10052 T^{5} + 118187 T^{6} - 308836 T^{7} + 2981470 T^{8} - 7168247 T^{9} + 61519486 T^{10} - 138610275 T^{11} + 1107057858 T^{12} - 138610275 p T^{13} + 61519486 p^{2} T^{14} - 7168247 p^{3} T^{15} + 2981470 p^{4} T^{16} - 308836 p^{5} T^{17} + 118187 p^{6} T^{18} - 10052 p^{7} T^{19} + 3652 p^{8} T^{20} - 189 p^{9} T^{21} + 83 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 34 T + 645 T^{2} + 8794 T^{3} + 95312 T^{4} + 865777 T^{5} + 6812069 T^{6} + 47474214 T^{7} + 297849242 T^{8} + 1702801198 T^{9} + 8951491634 T^{10} + 43548035213 T^{11} + 196815056450 T^{12} + 43548035213 p T^{13} + 8951491634 p^{2} T^{14} + 1702801198 p^{3} T^{15} + 297849242 p^{4} T^{16} + 47474214 p^{5} T^{17} + 6812069 p^{6} T^{18} + 865777 p^{7} T^{19} + 95312 p^{8} T^{20} + 8794 p^{9} T^{21} + 645 p^{10} T^{22} + 34 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 7 T + 191 T^{2} + 1377 T^{3} + 18324 T^{4} + 127923 T^{5} + 1153182 T^{6} + 7477418 T^{7} + 52331126 T^{8} + 307183151 T^{9} + 1786654975 T^{10} + 9340533422 T^{11} + 46788350498 T^{12} + 9340533422 p T^{13} + 1786654975 p^{2} T^{14} + 307183151 p^{3} T^{15} + 52331126 p^{4} T^{16} + 7477418 p^{5} T^{17} + 1153182 p^{6} T^{18} + 127923 p^{7} T^{19} + 18324 p^{8} T^{20} + 1377 p^{9} T^{21} + 191 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 6 T + 120 T^{2} + 14 p T^{3} + 7114 T^{4} + 16105 T^{5} + 316168 T^{6} + 424967 T^{7} + 11873587 T^{8} + 12616749 T^{9} + 420089582 T^{10} + 417148228 T^{11} + 12963011729 T^{12} + 417148228 p T^{13} + 420089582 p^{2} T^{14} + 12616749 p^{3} T^{15} + 11873587 p^{4} T^{16} + 424967 p^{5} T^{17} + 316168 p^{6} T^{18} + 16105 p^{7} T^{19} + 7114 p^{8} T^{20} + 14 p^{10} T^{21} + 120 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 52 T + 1487 T^{2} + 29804 T^{3} + 463898 T^{4} + 5917835 T^{5} + 64059196 T^{6} + 602812748 T^{7} + 5024057208 T^{8} + 37640854388 T^{9} + 256616314545 T^{10} + 1606849465779 T^{11} + 9297347180530 T^{12} + 1606849465779 p T^{13} + 256616314545 p^{2} T^{14} + 37640854388 p^{3} T^{15} + 5024057208 p^{4} T^{16} + 602812748 p^{5} T^{17} + 64059196 p^{6} T^{18} + 5917835 p^{7} T^{19} + 463898 p^{8} T^{20} + 29804 p^{9} T^{21} + 1487 p^{10} T^{22} + 52 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 3 T + 143 T^{2} - 377 T^{3} + 13205 T^{4} - 31844 T^{5} + 880199 T^{6} - 1931977 T^{7} + 47125840 T^{8} - 96525782 T^{9} + 2150817854 T^{10} - 4088427245 T^{11} + 84428899532 T^{12} - 4088427245 p T^{13} + 2150817854 p^{2} T^{14} - 96525782 p^{3} T^{15} + 47125840 p^{4} T^{16} - 1931977 p^{5} T^{17} + 880199 p^{6} T^{18} - 31844 p^{7} T^{19} + 13205 p^{8} T^{20} - 377 p^{9} T^{21} + 143 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 16 T + 343 T^{2} + 4525 T^{3} + 59386 T^{4} + 645720 T^{5} + 6627331 T^{6} + 61187664 T^{7} + 527665336 T^{8} + 4236442716 T^{9} + 31675100080 T^{10} + 223579922175 T^{11} + 1471202800985 T^{12} + 223579922175 p T^{13} + 31675100080 p^{2} T^{14} + 4236442716 p^{3} T^{15} + 527665336 p^{4} T^{16} + 61187664 p^{5} T^{17} + 6627331 p^{6} T^{18} + 645720 p^{7} T^{19} + 59386 p^{8} T^{20} + 4525 p^{9} T^{21} + 343 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 2 T + 265 T^{2} + 1357 T^{3} + 35615 T^{4} + 260762 T^{5} + 3497350 T^{6} + 644951 p T^{7} + 270649547 T^{8} + 2061792430 T^{9} + 16350518177 T^{10} + 116168135875 T^{11} + 783128133455 T^{12} + 116168135875 p T^{13} + 16350518177 p^{2} T^{14} + 2061792430 p^{3} T^{15} + 270649547 p^{4} T^{16} + 644951 p^{6} T^{17} + 3497350 p^{6} T^{18} + 260762 p^{7} T^{19} + 35615 p^{8} T^{20} + 1357 p^{9} T^{21} + 265 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 3 T + 250 T^{2} + 1337 T^{3} + 31552 T^{4} + 214897 T^{5} + 2858042 T^{6} + 19882190 T^{7} + 206049355 T^{8} + 1332661181 T^{9} + 12099304508 T^{10} + 73478633690 T^{11} + 605535774741 T^{12} + 73478633690 p T^{13} + 12099304508 p^{2} T^{14} + 1332661181 p^{3} T^{15} + 206049355 p^{4} T^{16} + 19882190 p^{5} T^{17} + 2858042 p^{6} T^{18} + 214897 p^{7} T^{19} + 31552 p^{8} T^{20} + 1337 p^{9} T^{21} + 250 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 19 T + 397 T^{2} - 4680 T^{3} + 62271 T^{4} - 558877 T^{5} + 5855591 T^{6} - 43278213 T^{7} + 405090680 T^{8} - 2648993320 T^{9} + 23815083656 T^{10} - 146202505875 T^{11} + 1300607366328 T^{12} - 146202505875 p T^{13} + 23815083656 p^{2} T^{14} - 2648993320 p^{3} T^{15} + 405090680 p^{4} T^{16} - 43278213 p^{5} T^{17} + 5855591 p^{6} T^{18} - 558877 p^{7} T^{19} + 62271 p^{8} T^{20} - 4680 p^{9} T^{21} + 397 p^{10} T^{22} - 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + T + 199 T^{2} - 287 T^{3} + 26260 T^{4} - 34024 T^{5} + 2604927 T^{6} - 4621952 T^{7} + 205127206 T^{8} - 342665585 T^{9} + 14643961590 T^{10} - 24968753201 T^{11} + 883987945058 T^{12} - 24968753201 p T^{13} + 14643961590 p^{2} T^{14} - 342665585 p^{3} T^{15} + 205127206 p^{4} T^{16} - 4621952 p^{5} T^{17} + 2604927 p^{6} T^{18} - 34024 p^{7} T^{19} + 26260 p^{8} T^{20} - 287 p^{9} T^{21} + 199 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 24 T + 739 T^{2} + 12958 T^{3} + 238844 T^{4} + 3337389 T^{5} + 46335329 T^{6} + 539258800 T^{7} + 6106510214 T^{8} + 60608324248 T^{9} + 580323067864 T^{10} + 4969281089561 T^{11} + 40920478743810 T^{12} + 4969281089561 p T^{13} + 580323067864 p^{2} T^{14} + 60608324248 p^{3} T^{15} + 6106510214 p^{4} T^{16} + 539258800 p^{5} T^{17} + 46335329 p^{6} T^{18} + 3337389 p^{7} T^{19} + 238844 p^{8} T^{20} + 12958 p^{9} T^{21} + 739 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 6 T + 474 T^{2} - 2801 T^{3} + 110233 T^{4} - 601273 T^{5} + 247293 p T^{6} - 80699047 T^{7} + 1813712476 T^{8} - 7816329367 T^{9} + 156738178491 T^{10} - 607428190822 T^{11} + 11341802615260 T^{12} - 607428190822 p T^{13} + 156738178491 p^{2} T^{14} - 7816329367 p^{3} T^{15} + 1813712476 p^{4} T^{16} - 80699047 p^{5} T^{17} + 247293 p^{7} T^{18} - 601273 p^{7} T^{19} + 110233 p^{8} T^{20} - 2801 p^{9} T^{21} + 474 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 15 T + 563 T^{2} + 7521 T^{3} + 160624 T^{4} + 1893692 T^{5} + 30102429 T^{6} + 313921184 T^{7} + 4087396800 T^{8} + 37859189305 T^{9} + 421622323828 T^{10} + 3472445873179 T^{11} + 33823440638886 T^{12} + 3472445873179 p T^{13} + 421622323828 p^{2} T^{14} + 37859189305 p^{3} T^{15} + 4087396800 p^{4} T^{16} + 313921184 p^{5} T^{17} + 30102429 p^{6} T^{18} + 1893692 p^{7} T^{19} + 160624 p^{8} T^{20} + 7521 p^{9} T^{21} + 563 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 20 T + 750 T^{2} + 12549 T^{3} + 266638 T^{4} + 3755257 T^{5} + 59005580 T^{6} + 708742678 T^{7} + 9039130426 T^{8} + 93766523279 T^{9} + 1011219935189 T^{10} + 9125134342013 T^{11} + 84870233902060 T^{12} + 9125134342013 p T^{13} + 1011219935189 p^{2} T^{14} + 93766523279 p^{3} T^{15} + 9039130426 p^{4} T^{16} + 708742678 p^{5} T^{17} + 59005580 p^{6} T^{18} + 3755257 p^{7} T^{19} + 266638 p^{8} T^{20} + 12549 p^{9} T^{21} + 750 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 53 T + 1876 T^{2} + 49594 T^{3} + 1085307 T^{4} + 20265867 T^{5} + 332914693 T^{6} + 4874352741 T^{7} + 64455524700 T^{8} + 774587682980 T^{9} + 8514401273847 T^{10} + 85824586349365 T^{11} + 795534726946528 T^{12} + 85824586349365 p T^{13} + 8514401273847 p^{2} T^{14} + 774587682980 p^{3} T^{15} + 64455524700 p^{4} T^{16} + 4874352741 p^{5} T^{17} + 332914693 p^{6} T^{18} + 20265867 p^{7} T^{19} + 1085307 p^{8} T^{20} + 49594 p^{9} T^{21} + 1876 p^{10} T^{22} + 53 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 17 T + 583 T^{2} - 7625 T^{3} + 158642 T^{4} - 1726174 T^{5} + 27757615 T^{6} - 263239654 T^{7} + 3603022534 T^{8} - 30730274277 T^{9} + 377415597750 T^{10} - 2963445605989 T^{11} + 33628996771766 T^{12} - 2963445605989 p T^{13} + 377415597750 p^{2} T^{14} - 30730274277 p^{3} T^{15} + 3603022534 p^{4} T^{16} - 263239654 p^{5} T^{17} + 27757615 p^{6} T^{18} - 1726174 p^{7} T^{19} + 158642 p^{8} T^{20} - 7625 p^{9} T^{21} + 583 p^{10} T^{22} - 17 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + T + 319 T^{2} + 622 T^{3} + 65659 T^{4} + 198965 T^{5} + 9408317 T^{6} + 42984890 T^{7} + 1074435416 T^{8} + 6554172363 T^{9} + 104754090189 T^{10} + 761297907629 T^{11} + 9511261379247 T^{12} + 761297907629 p T^{13} + 104754090189 p^{2} T^{14} + 6554172363 p^{3} T^{15} + 1074435416 p^{4} T^{16} + 42984890 p^{5} T^{17} + 9408317 p^{6} T^{18} + 198965 p^{7} T^{19} + 65659 p^{8} T^{20} + 622 p^{9} T^{21} + 319 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 12 T + 545 T^{2} - 4378 T^{3} + 129589 T^{4} - 725083 T^{5} + 20075299 T^{6} - 88886477 T^{7} + 2622570670 T^{8} - 11743295543 T^{9} + 318710454448 T^{10} - 1495390697603 T^{11} + 33862695620848 T^{12} - 1495390697603 p T^{13} + 318710454448 p^{2} T^{14} - 11743295543 p^{3} T^{15} + 2622570670 p^{4} T^{16} - 88886477 p^{5} T^{17} + 20075299 p^{6} T^{18} - 725083 p^{7} T^{19} + 129589 p^{8} T^{20} - 4378 p^{9} T^{21} + 545 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−4.29049695509124608662804672652, −4.20794305240342618901209651915, −4.16955528548761491998370361645, −3.99130097231142262541633055162, −3.94643229344038990604561753641, −3.74337864449531509357655909044, −3.74019648453374619114005095651, −3.72898829272516327044778624183, −3.70442910125322629449850042511, −3.63018628527675184387112084302, −3.52107567041324096239842505644, −3.33963619354482427844434602203, −3.28626201679494847329223716374, −2.93127496320463697870115769483, −2.91666537490191435358717896732, −2.88599904215601130944392137095, −2.70857959323148835340226762624, −2.58492994415211244658888385157, −2.43310312668450947584278987367, −2.22950151339791838756176820913, −2.21546867003659826213242284894, −2.11738266878372937996862453749, −2.05521596825884353780125603174, −1.52536752754547509875888279223, −1.51572372221591884380848142897, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.51572372221591884380848142897, 1.52536752754547509875888279223, 2.05521596825884353780125603174, 2.11738266878372937996862453749, 2.21546867003659826213242284894, 2.22950151339791838756176820913, 2.43310312668450947584278987367, 2.58492994415211244658888385157, 2.70857959323148835340226762624, 2.88599904215601130944392137095, 2.91666537490191435358717896732, 2.93127496320463697870115769483, 3.28626201679494847329223716374, 3.33963619354482427844434602203, 3.52107567041324096239842505644, 3.63018628527675184387112084302, 3.70442910125322629449850042511, 3.72898829272516327044778624183, 3.74019648453374619114005095651, 3.74337864449531509357655909044, 3.94643229344038990604561753641, 3.99130097231142262541633055162, 4.16955528548761491998370361645, 4.20794305240342618901209651915, 4.29049695509124608662804672652
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.