Dirichlet series
| L(s) = 1 | + 6·5-s − 18·7-s + 18·13-s + 6·17-s − 12·19-s + 18·23-s + 3·25-s + 2·27-s − 6·29-s − 18·31-s − 108·35-s + 6·41-s + 6·43-s − 6·47-s + 147·49-s − 36·53-s + 6·59-s − 6·61-s + 108·65-s − 6·67-s − 54·71-s − 12·73-s + 6·79-s − 18·83-s + 36·85-s + 24·89-s − 324·91-s + ⋯ |
| L(s) = 1 | + 2.68·5-s − 6.80·7-s + 4.99·13-s + 1.45·17-s − 2.75·19-s + 3.75·23-s + 3/5·25-s + 0.384·27-s − 1.11·29-s − 3.23·31-s − 18.2·35-s + 0.937·41-s + 0.914·43-s − 0.875·47-s + 21·49-s − 4.94·53-s + 0.781·59-s − 0.768·61-s + 13.3·65-s − 0.733·67-s − 6.40·71-s − 1.40·73-s + 0.675·79-s − 1.97·83-s + 3.90·85-s + 2.54·89-s − 33.9·91-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 3^{12} \cdot 19^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.22465\times 10^{10}\) |
| Root analytic conductor: | \(2.69858\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 3^{12} \cdot 19^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.9330889859\) |
| \(L(\frac12)\) | \(\approx\) | \(0.9330889859\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T^{3} + T^{6} )^{2} \) | |
| 19 | \( 1 + 12 T + 48 T^{2} - 14 T^{3} - 738 T^{4} - 2592 T^{5} - 6645 T^{6} - 2592 p T^{7} - 738 p^{2} T^{8} - 14 p^{3} T^{9} + 48 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 5 | \( 1 - 6 T + 33 T^{2} - 136 T^{3} + 96 p T^{4} - 1572 T^{5} + 4502 T^{6} - 2484 p T^{7} + 31203 T^{8} - 75844 T^{9} + 35643 p T^{10} - 402234 T^{11} + 921781 T^{12} - 402234 p T^{13} + 35643 p^{3} T^{14} - 75844 p^{3} T^{15} + 31203 p^{4} T^{16} - 2484 p^{6} T^{17} + 4502 p^{6} T^{18} - 1572 p^{7} T^{19} + 96 p^{9} T^{20} - 136 p^{9} T^{21} + 33 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 + 18 T + 177 T^{2} + 1242 T^{3} + 6852 T^{4} + 31482 T^{5} + 125871 T^{6} + 454248 T^{7} + 1526925 T^{8} + 697302 p T^{9} + 14916390 T^{10} + 6186564 p T^{11} + 118348673 T^{12} + 6186564 p^{2} T^{13} + 14916390 p^{2} T^{14} + 697302 p^{4} T^{15} + 1526925 p^{4} T^{16} + 454248 p^{5} T^{17} + 125871 p^{6} T^{18} + 31482 p^{7} T^{19} + 6852 p^{8} T^{20} + 1242 p^{9} T^{21} + 177 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 24 T^{2} + 360 T^{4} + 720 T^{5} + 2126 T^{6} + 9360 T^{7} - 5376 T^{8} + 69120 T^{9} - 7680 T^{10} - 675360 T^{11} - 911829 T^{12} - 675360 p T^{13} - 7680 p^{2} T^{14} + 69120 p^{3} T^{15} - 5376 p^{4} T^{16} + 9360 p^{5} T^{17} + 2126 p^{6} T^{18} + 720 p^{7} T^{19} + 360 p^{8} T^{20} + 24 p^{10} T^{22} + p^{12} T^{24} \) | |
| 13 | \( 1 - 18 T + 147 T^{2} - 756 T^{3} + 3294 T^{4} - 15516 T^{5} + 73524 T^{6} - 23832 p T^{7} + 7101 p^{2} T^{8} - 4546260 T^{9} + 17131629 T^{10} - 64347210 T^{11} + 236728961 T^{12} - 64347210 p T^{13} + 17131629 p^{2} T^{14} - 4546260 p^{3} T^{15} + 7101 p^{6} T^{16} - 23832 p^{6} T^{17} + 73524 p^{6} T^{18} - 15516 p^{7} T^{19} + 3294 p^{8} T^{20} - 756 p^{9} T^{21} + 147 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 6 T - 21 T^{2} + 116 T^{3} + 804 T^{4} - 4452 T^{5} - 6694 T^{6} + 3096 p T^{7} + 239895 T^{8} - 1176148 T^{9} - 1307523 T^{10} - 624462 T^{11} + 63960373 T^{12} - 624462 p T^{13} - 1307523 p^{2} T^{14} - 1176148 p^{3} T^{15} + 239895 p^{4} T^{16} + 3096 p^{6} T^{17} - 6694 p^{6} T^{18} - 4452 p^{7} T^{19} + 804 p^{8} T^{20} + 116 p^{9} T^{21} - 21 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 18 T + 249 T^{2} - 2664 T^{3} + 24510 T^{4} - 199152 T^{5} + 1470740 T^{6} - 9970056 T^{7} + 62934453 T^{8} - 371423700 T^{9} + 2061217359 T^{10} - 10780073082 T^{11} + 53204912097 T^{12} - 10780073082 p T^{13} + 2061217359 p^{2} T^{14} - 371423700 p^{3} T^{15} + 62934453 p^{4} T^{16} - 9970056 p^{5} T^{17} + 1470740 p^{6} T^{18} - 199152 p^{7} T^{19} + 24510 p^{8} T^{20} - 2664 p^{9} T^{21} + 249 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 6 T + 39 T^{2} - 420 T^{3} - 2400 T^{4} - 8148 T^{5} + 132188 T^{6} + 781704 T^{7} + 239199 T^{8} - 23753244 T^{9} - 129921195 T^{10} + 474804630 T^{11} + 4206760425 T^{12} + 474804630 p T^{13} - 129921195 p^{2} T^{14} - 23753244 p^{3} T^{15} + 239199 p^{4} T^{16} + 781704 p^{5} T^{17} + 132188 p^{6} T^{18} - 8148 p^{7} T^{19} - 2400 p^{8} T^{20} - 420 p^{9} T^{21} + 39 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 18 T + 129 T^{2} + 990 T^{3} + 10182 T^{4} + 66870 T^{5} + 432013 T^{6} + 3779676 T^{7} + 22837959 T^{8} + 120986838 T^{9} + 911021076 T^{10} + 5272786908 T^{11} + 24944461521 T^{12} + 5272786908 p T^{13} + 911021076 p^{2} T^{14} + 120986838 p^{3} T^{15} + 22837959 p^{4} T^{16} + 3779676 p^{5} T^{17} + 432013 p^{6} T^{18} + 66870 p^{7} T^{19} + 10182 p^{8} T^{20} + 990 p^{9} T^{21} + 129 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 270 T^{2} + 37461 T^{4} - 3454938 T^{6} + 234513930 T^{8} - 12294585414 T^{10} + 509770649165 T^{12} - 12294585414 p^{2} T^{14} + 234513930 p^{4} T^{16} - 3454938 p^{6} T^{18} + 37461 p^{8} T^{20} - 270 p^{10} T^{22} + p^{12} T^{24} \) | |
| 41 | \( 1 - 6 T + 93 T^{2} + 168 T^{3} - 2067 T^{4} + 48714 T^{5} - 145157 T^{6} - 509886 T^{7} + 13335930 T^{8} - 112676124 T^{9} - 12959460 T^{10} + 2302973082 T^{11} - 42267554175 T^{12} + 2302973082 p T^{13} - 12959460 p^{2} T^{14} - 112676124 p^{3} T^{15} + 13335930 p^{4} T^{16} - 509886 p^{5} T^{17} - 145157 p^{6} T^{18} + 48714 p^{7} T^{19} - 2067 p^{8} T^{20} + 168 p^{9} T^{21} + 93 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 6 T - 51 T^{2} + 456 T^{3} + 4245 T^{4} - 27894 T^{5} - 244693 T^{6} + 1307466 T^{7} + 13747716 T^{8} - 45465936 T^{9} - 639185004 T^{10} + 796038174 T^{11} + 29310587253 T^{12} + 796038174 p T^{13} - 639185004 p^{2} T^{14} - 45465936 p^{3} T^{15} + 13747716 p^{4} T^{16} + 1307466 p^{5} T^{17} - 244693 p^{6} T^{18} - 27894 p^{7} T^{19} + 4245 p^{8} T^{20} + 456 p^{9} T^{21} - 51 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 6 T - 21 T^{2} - 708 T^{3} - 9 T^{4} + 30714 T^{5} + 165899 T^{6} - 488178 T^{7} - 7430466 T^{8} - 27732936 T^{9} - 225549696 T^{10} + 1373003910 T^{11} + 9244716153 T^{12} + 1373003910 p T^{13} - 225549696 p^{2} T^{14} - 27732936 p^{3} T^{15} - 7430466 p^{4} T^{16} - 488178 p^{5} T^{17} + 165899 p^{6} T^{18} + 30714 p^{7} T^{19} - 9 p^{8} T^{20} - 708 p^{9} T^{21} - 21 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 36 T + 606 T^{2} + 5616 T^{3} + 21174 T^{4} - 169560 T^{5} - 3449098 T^{6} - 31814820 T^{7} - 205517880 T^{8} - 767720376 T^{9} + 4006486980 T^{10} + 107537288544 T^{11} + 1043615819619 T^{12} + 107537288544 p T^{13} + 4006486980 p^{2} T^{14} - 767720376 p^{3} T^{15} - 205517880 p^{4} T^{16} - 31814820 p^{5} T^{17} - 3449098 p^{6} T^{18} - 169560 p^{7} T^{19} + 21174 p^{8} T^{20} + 5616 p^{9} T^{21} + 606 p^{10} T^{22} + 36 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 6 T + 15 T^{2} - 28 T^{3} - 618 T^{4} - 13848 T^{5} - 144556 T^{6} - 574524 T^{7} + 10243197 T^{8} - 57664756 T^{9} + 335940441 T^{10} + 787901862 T^{11} + 43329031129 T^{12} + 787901862 p T^{13} + 335940441 p^{2} T^{14} - 57664756 p^{3} T^{15} + 10243197 p^{4} T^{16} - 574524 p^{5} T^{17} - 144556 p^{6} T^{18} - 13848 p^{7} T^{19} - 618 p^{8} T^{20} - 28 p^{9} T^{21} + 15 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 6 T - 21 T^{2} + 332 T^{3} + 6108 T^{4} - 4710 T^{5} - 87092 T^{6} + 47340 T^{7} + 4846374 T^{8} - 29617144 T^{9} - 1769529021 T^{10} - 9907277316 T^{11} + 29477640247 T^{12} - 9907277316 p T^{13} - 1769529021 p^{2} T^{14} - 29617144 p^{3} T^{15} + 4846374 p^{4} T^{16} + 47340 p^{5} T^{17} - 87092 p^{6} T^{18} - 4710 p^{7} T^{19} + 6108 p^{8} T^{20} + 332 p^{9} T^{21} - 21 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 6 T - 45 T^{2} + 284 T^{3} + 3132 T^{4} - 26004 T^{5} + 27304 T^{6} + 1796760 T^{7} - 8571933 T^{8} - 210455452 T^{9} + 483162465 T^{10} - 219933630 T^{11} - 175169010599 T^{12} - 219933630 p T^{13} + 483162465 p^{2} T^{14} - 210455452 p^{3} T^{15} - 8571933 p^{4} T^{16} + 1796760 p^{5} T^{17} + 27304 p^{6} T^{18} - 26004 p^{7} T^{19} + 3132 p^{8} T^{20} + 284 p^{9} T^{21} - 45 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( ( 1 + 27 T + 369 T^{2} + 3379 T^{3} + 31932 T^{4} + 363186 T^{5} + 3587021 T^{6} + 363186 p T^{7} + 31932 p^{2} T^{8} + 3379 p^{3} T^{9} + 369 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 73 | \( 1 + 12 T + 264 T^{2} + 1614 T^{3} + 29730 T^{4} + 139314 T^{5} + 3188443 T^{6} + 14988528 T^{7} + 331334937 T^{8} + 1490563710 T^{9} + 27741144429 T^{10} + 1322660826 p T^{11} + 26543588361 p T^{12} + 1322660826 p^{2} T^{13} + 27741144429 p^{2} T^{14} + 1490563710 p^{3} T^{15} + 331334937 p^{4} T^{16} + 14988528 p^{5} T^{17} + 3188443 p^{6} T^{18} + 139314 p^{7} T^{19} + 29730 p^{8} T^{20} + 1614 p^{9} T^{21} + 264 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 6 T - 111 T^{2} + 1268 T^{3} - 2379 T^{4} - 20562 T^{5} + 909943 T^{6} - 5230458 T^{7} - 9967086 T^{8} + 97426772 T^{9} - 903017748 T^{10} + 17978764890 T^{11} + 13591474441 T^{12} + 17978764890 p T^{13} - 903017748 p^{2} T^{14} + 97426772 p^{3} T^{15} - 9967086 p^{4} T^{16} - 5230458 p^{5} T^{17} + 909943 p^{6} T^{18} - 20562 p^{7} T^{19} - 2379 p^{8} T^{20} + 1268 p^{9} T^{21} - 111 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 18 T + 327 T^{2} + 3942 T^{3} + 41448 T^{4} + 377442 T^{5} + 2442793 T^{6} + 12026232 T^{7} - 33567381 T^{8} - 1296043362 T^{9} - 19312803432 T^{10} - 221448999492 T^{11} - 2132569291431 T^{12} - 221448999492 p T^{13} - 19312803432 p^{2} T^{14} - 1296043362 p^{3} T^{15} - 33567381 p^{4} T^{16} + 12026232 p^{5} T^{17} + 2442793 p^{6} T^{18} + 377442 p^{7} T^{19} + 41448 p^{8} T^{20} + 3942 p^{9} T^{21} + 327 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 24 T + 6 T^{2} + 8574 T^{3} - 124668 T^{4} - 294954 T^{5} + 28385759 T^{6} - 273103704 T^{7} - 1326387117 T^{8} + 50528178066 T^{9} - 325217881443 T^{10} - 2503064412918 T^{11} + 53965929184197 T^{12} - 2503064412918 p T^{13} - 325217881443 p^{2} T^{14} + 50528178066 p^{3} T^{15} - 1326387117 p^{4} T^{16} - 273103704 p^{5} T^{17} + 28385759 p^{6} T^{18} - 294954 p^{7} T^{19} - 124668 p^{8} T^{20} + 8574 p^{9} T^{21} + 6 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 24 T + 78 T^{2} + 1122 T^{3} - 2469 T^{4} + 2316 p T^{5} - 2900109 T^{6} - 31359102 T^{7} + 364826805 T^{8} - 242240634 T^{9} + 26053983087 T^{10} + 20042013666 T^{11} - 5243177200642 T^{12} + 20042013666 p T^{13} + 26053983087 p^{2} T^{14} - 242240634 p^{3} T^{15} + 364826805 p^{4} T^{16} - 31359102 p^{5} T^{17} - 2900109 p^{6} T^{18} + 2316 p^{8} T^{19} - 2469 p^{8} T^{20} + 1122 p^{9} T^{21} + 78 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.22585796092145700851537711404, −3.16905716014193750962228608368, −3.06555608110830367714452421109, −3.02387903717406894075678338598, −3.01838501257382522691816972240, −2.82963609415281371227128277293, −2.82947526906145162424626551254, −2.57863914347404591773050374175, −2.54370175392310692467677788682, −2.33673039155900585335964922026, −2.21944429611909830790279528517, −1.91375420997395028909550144153, −1.90726886021590743794304058600, −1.84215918807774164847718353128, −1.80478940238336796505629930019, −1.80340510327240565857055078674, −1.63285273148779585081730148740, −1.19691084887354028961865104010, −1.19567756697530893877107556320, −1.17926299223077605496097583936, −1.06997527538100243579203997051, −0.839332840279782550439219896923, −0.30999821381403498912990888687, −0.24096784396713839048231158020, −0.21930207734838549589881648597, 0.21930207734838549589881648597, 0.24096784396713839048231158020, 0.30999821381403498912990888687, 0.839332840279782550439219896923, 1.06997527538100243579203997051, 1.17926299223077605496097583936, 1.19567756697530893877107556320, 1.19691084887354028961865104010, 1.63285273148779585081730148740, 1.80340510327240565857055078674, 1.80478940238336796505629930019, 1.84215918807774164847718353128, 1.90726886021590743794304058600, 1.91375420997395028909550144153, 2.21944429611909830790279528517, 2.33673039155900585335964922026, 2.54370175392310692467677788682, 2.57863914347404591773050374175, 2.82947526906145162424626551254, 2.82963609415281371227128277293, 3.01838501257382522691816972240, 3.02387903717406894075678338598, 3.06555608110830367714452421109, 3.16905716014193750962228608368, 3.22585796092145700851537711404
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.