Dirichlet series
| L(s) = 1 | + 3·3-s + 7·5-s + 2·7-s − 18·11-s − 4·13-s + 21·15-s + 12·17-s + 6·21-s + 9·23-s + 7·25-s − 6·27-s + 34·29-s + 11·31-s − 54·33-s + 14·35-s − 22·37-s − 12·39-s + 32·41-s + 8·43-s − 24·47-s − 22·49-s + 36·51-s − 2·53-s − 126·55-s − 26·59-s + 12·61-s − 28·65-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 3.13·5-s + 0.755·7-s − 5.42·11-s − 1.10·13-s + 5.42·15-s + 2.91·17-s + 1.30·21-s + 1.87·23-s + 7/5·25-s − 1.15·27-s + 6.31·29-s + 1.97·31-s − 9.40·33-s + 2.36·35-s − 3.61·37-s − 1.92·39-s + 4.99·41-s + 1.21·43-s − 3.50·47-s − 3.14·49-s + 5.04·51-s − 0.274·53-s − 16.9·55-s − 3.38·59-s + 1.53·61-s − 3.47·65-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 503^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 503^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 503^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.96113\times 10^{21}\) |
| Root analytic conductor: | \(8.01645\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 503^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(183.1530757\) |
| \(L(\frac12)\) | \(\approx\) | \(183.1530757\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 503 | \( ( 1 + T )^{12} \) | |
| good | 3 | \( 1 - p T + p^{2} T^{2} - 7 p T^{3} + 17 p T^{4} - 125 T^{5} + 293 T^{6} - 592 T^{7} + 1250 T^{8} - 2291 T^{9} + 4498 T^{10} - 8234 T^{11} + 4876 p T^{12} - 8234 p T^{13} + 4498 p^{2} T^{14} - 2291 p^{3} T^{15} + 1250 p^{4} T^{16} - 592 p^{5} T^{17} + 293 p^{6} T^{18} - 125 p^{7} T^{19} + 17 p^{9} T^{20} - 7 p^{10} T^{21} + p^{12} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 - 7 T + 42 T^{2} - 38 p T^{3} + 781 T^{4} - 2772 T^{5} + 9074 T^{6} - 5438 p T^{7} + 76806 T^{8} - 40498 p T^{9} + 509798 T^{10} - 1215619 T^{11} + 2791764 T^{12} - 1215619 p T^{13} + 509798 p^{2} T^{14} - 40498 p^{4} T^{15} + 76806 p^{4} T^{16} - 5438 p^{6} T^{17} + 9074 p^{6} T^{18} - 2772 p^{7} T^{19} + 781 p^{8} T^{20} - 38 p^{10} T^{21} + 42 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 - 2 T + 26 T^{2} - 41 T^{3} + 387 T^{4} - 495 T^{5} + 4456 T^{6} - 690 p T^{7} + 43901 T^{8} - 43875 T^{9} + 380095 T^{10} - 51587 p T^{11} + 408629 p T^{12} - 51587 p^{2} T^{13} + 380095 p^{2} T^{14} - 43875 p^{3} T^{15} + 43901 p^{4} T^{16} - 690 p^{6} T^{17} + 4456 p^{6} T^{18} - 495 p^{7} T^{19} + 387 p^{8} T^{20} - 41 p^{9} T^{21} + 26 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 18 T + 206 T^{2} + 1751 T^{3} + 12433 T^{4} + 76100 T^{5} + 416305 T^{6} + 2060807 T^{7} + 9385744 T^{8} + 39508541 T^{9} + 155009377 T^{10} + 4689169 p^{2} T^{11} + 1945839084 T^{12} + 4689169 p^{3} T^{13} + 155009377 p^{2} T^{14} + 39508541 p^{3} T^{15} + 9385744 p^{4} T^{16} + 2060807 p^{5} T^{17} + 416305 p^{6} T^{18} + 76100 p^{7} T^{19} + 12433 p^{8} T^{20} + 1751 p^{9} T^{21} + 206 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 4 T + 5 p T^{2} + 248 T^{3} + 2225 T^{4} + 609 p T^{5} + 54689 T^{6} + 179986 T^{7} + 82560 p T^{8} + 3265684 T^{9} + 17414630 T^{10} + 49451901 T^{11} + 242072156 T^{12} + 49451901 p T^{13} + 17414630 p^{2} T^{14} + 3265684 p^{3} T^{15} + 82560 p^{5} T^{16} + 179986 p^{5} T^{17} + 54689 p^{6} T^{18} + 609 p^{8} T^{19} + 2225 p^{8} T^{20} + 248 p^{9} T^{21} + 5 p^{11} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 12 T + 166 T^{2} - 1221 T^{3} + 9798 T^{4} - 51318 T^{5} + 17554 p T^{6} - 1123183 T^{7} + 5047911 T^{8} - 11448777 T^{9} + 42672488 T^{10} + 1695375 T^{11} + 253046308 T^{12} + 1695375 p T^{13} + 42672488 p^{2} T^{14} - 11448777 p^{3} T^{15} + 5047911 p^{4} T^{16} - 1123183 p^{5} T^{17} + 17554 p^{7} T^{18} - 51318 p^{7} T^{19} + 9798 p^{8} T^{20} - 1221 p^{9} T^{21} + 166 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 59 T^{2} + 67 T^{3} + 2447 T^{4} + 3079 T^{5} + 74847 T^{6} + 136062 T^{7} + 1860811 T^{8} + 3815153 T^{9} + 41316958 T^{10} + 91606659 T^{11} + 798996362 T^{12} + 91606659 p T^{13} + 41316958 p^{2} T^{14} + 3815153 p^{3} T^{15} + 1860811 p^{4} T^{16} + 136062 p^{5} T^{17} + 74847 p^{6} T^{18} + 3079 p^{7} T^{19} + 2447 p^{8} T^{20} + 67 p^{9} T^{21} + 59 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( 1 - 9 T + 157 T^{2} - 1198 T^{3} + 12814 T^{4} - 84379 T^{5} + 700261 T^{6} - 4074957 T^{7} + 28422478 T^{8} - 148320623 T^{9} + 904244314 T^{10} - 4251116175 T^{11} + 23114338603 T^{12} - 4251116175 p T^{13} + 904244314 p^{2} T^{14} - 148320623 p^{3} T^{15} + 28422478 p^{4} T^{16} - 4074957 p^{5} T^{17} + 700261 p^{6} T^{18} - 84379 p^{7} T^{19} + 12814 p^{8} T^{20} - 1198 p^{9} T^{21} + 157 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 34 T + 717 T^{2} - 11125 T^{3} + 141257 T^{4} - 1531538 T^{5} + 14628117 T^{6} - 125171225 T^{7} + 971974667 T^{8} - 6905339085 T^{9} + 45156961894 T^{10} - 272841031757 T^{11} + 1526411184342 T^{12} - 272841031757 p T^{13} + 45156961894 p^{2} T^{14} - 6905339085 p^{3} T^{15} + 971974667 p^{4} T^{16} - 125171225 p^{5} T^{17} + 14628117 p^{6} T^{18} - 1531538 p^{7} T^{19} + 141257 p^{8} T^{20} - 11125 p^{9} T^{21} + 717 p^{10} T^{22} - 34 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 11 T + 246 T^{2} - 2242 T^{3} + 28987 T^{4} - 229197 T^{5} + 2208928 T^{6} - 15528844 T^{7} + 122865191 T^{8} - 777995717 T^{9} + 5304569738 T^{10} - 30338429999 T^{11} + 183095285754 T^{12} - 30338429999 p T^{13} + 5304569738 p^{2} T^{14} - 777995717 p^{3} T^{15} + 122865191 p^{4} T^{16} - 15528844 p^{5} T^{17} + 2208928 p^{6} T^{18} - 229197 p^{7} T^{19} + 28987 p^{8} T^{20} - 2242 p^{9} T^{21} + 246 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 22 T + 455 T^{2} + 6559 T^{3} + 85475 T^{4} + 942209 T^{5} + 9511859 T^{6} + 86045010 T^{7} + 722793987 T^{8} + 5570654269 T^{9} + 1087621842 p T^{10} + 269557457559 T^{11} + 1700095666794 T^{12} + 269557457559 p T^{13} + 1087621842 p^{3} T^{14} + 5570654269 p^{3} T^{15} + 722793987 p^{4} T^{16} + 86045010 p^{5} T^{17} + 9511859 p^{6} T^{18} + 942209 p^{7} T^{19} + 85475 p^{8} T^{20} + 6559 p^{9} T^{21} + 455 p^{10} T^{22} + 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 32 T + 770 T^{2} - 13419 T^{3} + 199542 T^{4} - 2512554 T^{5} + 28288052 T^{6} - 283801421 T^{7} + 2605368715 T^{8} - 21811184839 T^{9} + 169130978058 T^{10} - 1209112436159 T^{11} + 8049974165820 T^{12} - 1209112436159 p T^{13} + 169130978058 p^{2} T^{14} - 21811184839 p^{3} T^{15} + 2605368715 p^{4} T^{16} - 283801421 p^{5} T^{17} + 28288052 p^{6} T^{18} - 2512554 p^{7} T^{19} + 199542 p^{8} T^{20} - 13419 p^{9} T^{21} + 770 p^{10} T^{22} - 32 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 8 T + 213 T^{2} - 1770 T^{3} + 23231 T^{4} - 166041 T^{5} + 1560689 T^{6} - 8765026 T^{7} + 63390606 T^{8} - 259839552 T^{9} + 1507551598 T^{10} - 3483878321 T^{11} + 34544523436 T^{12} - 3483878321 p T^{13} + 1507551598 p^{2} T^{14} - 259839552 p^{3} T^{15} + 63390606 p^{4} T^{16} - 8765026 p^{5} T^{17} + 1560689 p^{6} T^{18} - 166041 p^{7} T^{19} + 23231 p^{8} T^{20} - 1770 p^{9} T^{21} + 213 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 24 T + 624 T^{2} + 9813 T^{3} + 154353 T^{4} + 1867001 T^{5} + 474176 p T^{6} + 222278140 T^{7} + 2182329723 T^{8} + 18581477729 T^{9} + 155728331067 T^{10} + 1151472566375 T^{11} + 8382970589145 T^{12} + 1151472566375 p T^{13} + 155728331067 p^{2} T^{14} + 18581477729 p^{3} T^{15} + 2182329723 p^{4} T^{16} + 222278140 p^{5} T^{17} + 474176 p^{7} T^{18} + 1867001 p^{7} T^{19} + 154353 p^{8} T^{20} + 9813 p^{9} T^{21} + 624 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 2 T + 395 T^{2} + 573 T^{3} + 79159 T^{4} + 85111 T^{5} + 10523539 T^{6} + 8593086 T^{7} + 1028184511 T^{8} + 664497415 T^{9} + 77483123342 T^{10} + 41921174393 T^{11} + 4607531907962 T^{12} + 41921174393 p T^{13} + 77483123342 p^{2} T^{14} + 664497415 p^{3} T^{15} + 1028184511 p^{4} T^{16} + 8593086 p^{5} T^{17} + 10523539 p^{6} T^{18} + 85111 p^{7} T^{19} + 79159 p^{8} T^{20} + 573 p^{9} T^{21} + 395 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 26 T + 437 T^{2} + 5246 T^{3} + 56014 T^{4} + 561929 T^{5} + 5627753 T^{6} + 53428741 T^{7} + 487722895 T^{8} + 4192620027 T^{9} + 35209729306 T^{10} + 283290463699 T^{11} + 2227511236260 T^{12} + 283290463699 p T^{13} + 35209729306 p^{2} T^{14} + 4192620027 p^{3} T^{15} + 487722895 p^{4} T^{16} + 53428741 p^{5} T^{17} + 5627753 p^{6} T^{18} + 561929 p^{7} T^{19} + 56014 p^{8} T^{20} + 5246 p^{9} T^{21} + 437 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 12 T + 461 T^{2} - 4767 T^{3} + 106261 T^{4} - 975590 T^{5} + 16121299 T^{6} - 133141815 T^{7} + 1794097610 T^{8} - 13394248577 T^{9} + 154375149996 T^{10} - 1039460652715 T^{11} + 10534585951880 T^{12} - 1039460652715 p T^{13} + 154375149996 p^{2} T^{14} - 13394248577 p^{3} T^{15} + 1794097610 p^{4} T^{16} - 133141815 p^{5} T^{17} + 16121299 p^{6} T^{18} - 975590 p^{7} T^{19} + 106261 p^{8} T^{20} - 4767 p^{9} T^{21} + 461 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 21 T + 458 T^{2} - 6321 T^{3} + 87583 T^{4} - 965771 T^{5} + 10922727 T^{6} - 107004290 T^{7} + 1094708152 T^{8} - 9997299639 T^{9} + 94636033095 T^{10} - 802402093488 T^{11} + 6948793830000 T^{12} - 802402093488 p T^{13} + 94636033095 p^{2} T^{14} - 9997299639 p^{3} T^{15} + 1094708152 p^{4} T^{16} - 107004290 p^{5} T^{17} + 10922727 p^{6} T^{18} - 965771 p^{7} T^{19} + 87583 p^{8} T^{20} - 6321 p^{9} T^{21} + 458 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 50 T + 1711 T^{2} + 43305 T^{3} + 908171 T^{4} + 16189750 T^{5} + 253701007 T^{6} + 3535881037 T^{7} + 44474886435 T^{8} + 507739217863 T^{9} + 5299361943762 T^{10} + 50676244010449 T^{11} + 445479921732642 T^{12} + 50676244010449 p T^{13} + 5299361943762 p^{2} T^{14} + 507739217863 p^{3} T^{15} + 44474886435 p^{4} T^{16} + 3535881037 p^{5} T^{17} + 253701007 p^{6} T^{18} + 16189750 p^{7} T^{19} + 908171 p^{8} T^{20} + 43305 p^{9} T^{21} + 1711 p^{10} T^{22} + 50 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 17 T + 610 T^{2} - 8290 T^{3} + 174647 T^{4} - 2005546 T^{5} + 31885540 T^{6} - 318801846 T^{7} + 4202591870 T^{8} - 37297650102 T^{9} + 427206769000 T^{10} - 3402453895951 T^{11} + 34693832766584 T^{12} - 3402453895951 p T^{13} + 427206769000 p^{2} T^{14} - 37297650102 p^{3} T^{15} + 4202591870 p^{4} T^{16} - 318801846 p^{5} T^{17} + 31885540 p^{6} T^{18} - 2005546 p^{7} T^{19} + 174647 p^{8} T^{20} - 8290 p^{9} T^{21} + 610 p^{10} T^{22} - 17 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 9 T + 648 T^{2} - 6279 T^{3} + 2594 p T^{4} - 2056730 T^{5} + 42044625 T^{6} - 5327328 p T^{7} + 6250252261 T^{8} - 60111649704 T^{9} + 709782021291 T^{10} - 6315539448032 T^{11} + 63127054087280 T^{12} - 6315539448032 p T^{13} + 709782021291 p^{2} T^{14} - 60111649704 p^{3} T^{15} + 6250252261 p^{4} T^{16} - 5327328 p^{6} T^{17} + 42044625 p^{6} T^{18} - 2056730 p^{7} T^{19} + 2594 p^{9} T^{20} - 6279 p^{9} T^{21} + 648 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 25 T + 697 T^{2} + 12233 T^{3} + 208949 T^{4} + 2854575 T^{5} + 37114827 T^{6} + 419442950 T^{7} + 4525988754 T^{8} + 44654078439 T^{9} + 428913306760 T^{10} + 3945221485244 T^{11} + 36292751560520 T^{12} + 3945221485244 p T^{13} + 428913306760 p^{2} T^{14} + 44654078439 p^{3} T^{15} + 4525988754 p^{4} T^{16} + 419442950 p^{5} T^{17} + 37114827 p^{6} T^{18} + 2854575 p^{7} T^{19} + 208949 p^{8} T^{20} + 12233 p^{9} T^{21} + 697 p^{10} T^{22} + 25 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 21 T + 781 T^{2} - 13992 T^{3} + 297410 T^{4} - 4532053 T^{5} + 72309901 T^{6} - 948480268 T^{7} + 12500242567 T^{8} - 143122686371 T^{9} + 1625460925398 T^{10} - 16402737491831 T^{11} + 163663794940716 T^{12} - 16402737491831 p T^{13} + 1625460925398 p^{2} T^{14} - 143122686371 p^{3} T^{15} + 12500242567 p^{4} T^{16} - 948480268 p^{5} T^{17} + 72309901 p^{6} T^{18} - 4532053 p^{7} T^{19} + 297410 p^{8} T^{20} - 13992 p^{9} T^{21} + 781 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 18 T + 591 T^{2} - 8155 T^{3} + 166085 T^{4} - 1930089 T^{5} + 31869251 T^{6} - 331651870 T^{7} + 4795815957 T^{8} - 45773961233 T^{9} + 596253361156 T^{10} - 5240536538331 T^{11} + 62648823242718 T^{12} - 5240536538331 p T^{13} + 596253361156 p^{2} T^{14} - 45773961233 p^{3} T^{15} + 4795815957 p^{4} T^{16} - 331651870 p^{5} T^{17} + 31869251 p^{6} T^{18} - 1930089 p^{7} T^{19} + 166085 p^{8} T^{20} - 8155 p^{9} T^{21} + 591 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.46487169572607320888232877369, −2.27573919513628334756701405668, −1.96513154866873135129844263583, −1.94418313569977646601044979294, −1.92408235764200021903851922547, −1.91763725215237820258556204599, −1.91297503381265311700155691606, −1.87295983419753978885013241519, −1.81264815258404985951696011707, −1.79149064337096230250465729803, −1.49379065467643381006761546385, −1.42298042129291639795112181588, −1.41182677897770759513353144237, −1.36121258065766087240366763252, −1.09623280273630181657976267979, −1.07556479230360626025057832723, −0.892432236697359087737479574598, −0.845551803525540769009745422640, −0.793079346389414657106718225030, −0.67891750404729943616230164604, −0.51724157665433537752669279547, −0.51516119786425573527353651150, −0.32796274955144405769035736102, −0.31167852269659222745502786391, −0.21404784451318173930920510098, 0.21404784451318173930920510098, 0.31167852269659222745502786391, 0.32796274955144405769035736102, 0.51516119786425573527353651150, 0.51724157665433537752669279547, 0.67891750404729943616230164604, 0.793079346389414657106718225030, 0.845551803525540769009745422640, 0.892432236697359087737479574598, 1.07556479230360626025057832723, 1.09623280273630181657976267979, 1.36121258065766087240366763252, 1.41182677897770759513353144237, 1.42298042129291639795112181588, 1.49379065467643381006761546385, 1.79149064337096230250465729803, 1.81264815258404985951696011707, 1.87295983419753978885013241519, 1.91297503381265311700155691606, 1.91763725215237820258556204599, 1.92408235764200021903851922547, 1.94418313569977646601044979294, 1.96513154866873135129844263583, 2.27573919513628334756701405668, 2.46487169572607320888232877369
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.