Dirichlet series
| L(s) = 1 | + 12·2-s + 2·3-s + 78·4-s + 6·5-s + 24·6-s + 8·7-s + 364·8-s + 72·10-s − 7·11-s + 156·12-s + 2·13-s + 96·14-s + 12·15-s + 1.36e3·16-s + 7·17-s + 14·19-s + 468·20-s + 16·21-s − 84·22-s − 9·23-s + 728·24-s + 15·25-s + 24·26-s + 27-s + 624·28-s − 9·29-s + 144·30-s + ⋯ |
| L(s) = 1 | + 8.48·2-s + 1.15·3-s + 39·4-s + 2.68·5-s + 9.79·6-s + 3.02·7-s + 128.·8-s + 22.7·10-s − 2.11·11-s + 45.0·12-s + 0.554·13-s + 25.6·14-s + 3.09·15-s + 341.·16-s + 1.69·17-s + 3.21·19-s + 104.·20-s + 3.49·21-s − 17.9·22-s − 1.87·23-s + 148.·24-s + 3·25-s + 4.70·26-s + 0.192·27-s + 117.·28-s − 1.67·29-s + 26.2·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{24} \cdot 5^{12} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.62670\times 10^{8}\) |
| Root analytic conductor: | \(2.24289\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{24} \cdot 5^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(30309.09072\) |
| \(L(\frac12)\) | \(\approx\) | \(30309.09072\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 - 2 T + 4 T^{2} - p^{2} T^{3} + 5 p T^{4} - 13 p T^{5} + 25 p T^{6} - 13 p^{2} T^{7} + 5 p^{3} T^{8} - p^{5} T^{9} + 4 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) | |
| 5 | \( ( 1 - T + T^{2} )^{6} \) | |
| 7 | \( ( 1 - 4 T + 2 p T^{2} - 55 T^{3} + 2 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | |
| good | 11 | \( 1 + 7 T - 5 T^{2} - 100 T^{3} + 96 T^{4} + 824 T^{5} - 2267 T^{6} - 5202 T^{7} + 18548 T^{8} - 3791 T^{9} - 96786 T^{10} + 3932 p^{2} T^{11} + 245275 p T^{12} + 3932 p^{3} T^{13} - 96786 p^{2} T^{14} - 3791 p^{3} T^{15} + 18548 p^{4} T^{16} - 5202 p^{5} T^{17} - 2267 p^{6} T^{18} + 824 p^{7} T^{19} + 96 p^{8} T^{20} - 100 p^{9} T^{21} - 5 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) |
| 13 | \( 1 - 2 T - 36 T^{3} - 17 T^{4} - 371 T^{5} + 3240 T^{6} - 7131 T^{7} + 28438 T^{8} - 222777 T^{9} + 460892 T^{10} - 507411 T^{11} + 7449937 T^{12} - 507411 p T^{13} + 460892 p^{2} T^{14} - 222777 p^{3} T^{15} + 28438 p^{4} T^{16} - 7131 p^{5} T^{17} + 3240 p^{6} T^{18} - 371 p^{7} T^{19} - 17 p^{8} T^{20} - 36 p^{9} T^{21} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 7 T - 26 T^{2} + 253 T^{3} + 159 T^{4} - 2042 T^{5} - 15857 T^{6} + 20745 T^{7} + 626078 T^{8} - 1325347 T^{9} - 9319905 T^{10} + 15360616 T^{11} + 109941593 T^{12} + 15360616 p T^{13} - 9319905 p^{2} T^{14} - 1325347 p^{3} T^{15} + 626078 p^{4} T^{16} + 20745 p^{5} T^{17} - 15857 p^{6} T^{18} - 2042 p^{7} T^{19} + 159 p^{8} T^{20} + 253 p^{9} T^{21} - 26 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 14 T + 66 T^{2} - 444 T^{3} + 4717 T^{4} - 22391 T^{5} + 98142 T^{6} - 823887 T^{7} + 3750856 T^{8} - 13240737 T^{9} + 95742878 T^{10} - 421689765 T^{11} + 1264726999 T^{12} - 421689765 p T^{13} + 95742878 p^{2} T^{14} - 13240737 p^{3} T^{15} + 3750856 p^{4} T^{16} - 823887 p^{5} T^{17} + 98142 p^{6} T^{18} - 22391 p^{7} T^{19} + 4717 p^{8} T^{20} - 444 p^{9} T^{21} + 66 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 9 T - 24 T^{2} - 45 T^{3} + 3480 T^{4} + 4437 T^{5} - 43177 T^{6} + 508653 T^{7} + 1537083 T^{8} - 6790500 T^{9} + 64602594 T^{10} + 262176318 T^{11} - 931869183 T^{12} + 262176318 p T^{13} + 64602594 p^{2} T^{14} - 6790500 p^{3} T^{15} + 1537083 p^{4} T^{16} + 508653 p^{5} T^{17} - 43177 p^{6} T^{18} + 4437 p^{7} T^{19} + 3480 p^{8} T^{20} - 45 p^{9} T^{21} - 24 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 9 T - 69 T^{2} - 702 T^{3} + 3759 T^{4} + 34596 T^{5} - 143518 T^{6} - 1156437 T^{7} + 4050324 T^{8} + 25653105 T^{9} - 98719569 T^{10} - 302648535 T^{11} + 2262044085 T^{12} - 302648535 p T^{13} - 98719569 p^{2} T^{14} + 25653105 p^{3} T^{15} + 4050324 p^{4} T^{16} - 1156437 p^{5} T^{17} - 143518 p^{6} T^{18} + 34596 p^{7} T^{19} + 3759 p^{8} T^{20} - 702 p^{9} T^{21} - 69 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( ( 1 + 9 T + 96 T^{2} + 767 T^{3} + 5613 T^{4} + 34584 T^{5} + 205869 T^{6} + 34584 p T^{7} + 5613 p^{2} T^{8} + 767 p^{3} T^{9} + 96 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( 1 + 12 T - 75 T^{2} - 1124 T^{3} + 5793 T^{4} + 66639 T^{5} - 404162 T^{6} - 2798919 T^{7} + 24121197 T^{8} + 87063469 T^{9} - 1138257942 T^{10} - 36115437 p T^{11} + 44567264293 T^{12} - 36115437 p^{2} T^{13} - 1138257942 p^{2} T^{14} + 87063469 p^{3} T^{15} + 24121197 p^{4} T^{16} - 2798919 p^{5} T^{17} - 404162 p^{6} T^{18} + 66639 p^{7} T^{19} + 5793 p^{8} T^{20} - 1124 p^{9} T^{21} - 75 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - T - 197 T^{2} + 250 T^{3} + 21102 T^{4} - 27179 T^{5} - 1633109 T^{6} + 1654722 T^{7} + 101292122 T^{8} - 62228170 T^{9} - 5261216175 T^{10} + 1034229106 T^{11} + 233114446817 T^{12} + 1034229106 p T^{13} - 5261216175 p^{2} T^{14} - 62228170 p^{3} T^{15} + 101292122 p^{4} T^{16} + 1654722 p^{5} T^{17} - 1633109 p^{6} T^{18} - 27179 p^{7} T^{19} + 21102 p^{8} T^{20} + 250 p^{9} T^{21} - 197 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 7 T - 61 T^{2} - 38 T^{3} + 2916 T^{4} + 23601 T^{5} + 1081 T^{6} - 108892 T^{7} - 5446372 T^{8} - 63105082 T^{9} - 148348975 T^{10} + 1479379556 T^{11} + 27501185741 T^{12} + 1479379556 p T^{13} - 148348975 p^{2} T^{14} - 63105082 p^{3} T^{15} - 5446372 p^{4} T^{16} - 108892 p^{5} T^{17} + 1081 p^{6} T^{18} + 23601 p^{7} T^{19} + 2916 p^{8} T^{20} - 38 p^{9} T^{21} - 61 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( ( 1 + 7 T + 192 T^{2} + 1490 T^{3} + 18046 T^{4} + 130951 T^{5} + 1054511 T^{6} + 130951 p T^{7} + 18046 p^{2} T^{8} + 1490 p^{3} T^{9} + 192 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 53 | \( 1 - 2 T - 167 T^{2} + 1010 T^{3} + 13227 T^{4} - 131245 T^{5} - 429158 T^{6} + 9790659 T^{7} - 11991223 T^{8} - 447494225 T^{9} + 2419646052 T^{10} + 9239703395 T^{11} - 165502278643 T^{12} + 9239703395 p T^{13} + 2419646052 p^{2} T^{14} - 447494225 p^{3} T^{15} - 11991223 p^{4} T^{16} + 9790659 p^{5} T^{17} - 429158 p^{6} T^{18} - 131245 p^{7} T^{19} + 13227 p^{8} T^{20} + 1010 p^{9} T^{21} - 167 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( ( 1 + 29 T + 597 T^{2} + 8413 T^{3} + 99691 T^{4} + 952679 T^{5} + 8015939 T^{6} + 952679 p T^{7} + 99691 p^{2} T^{8} + 8413 p^{3} T^{9} + 597 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 61 | \( ( 1 - 11 T + 362 T^{2} - 3015 T^{3} + 54025 T^{4} - 347434 T^{5} + 4351741 T^{6} - 347434 p T^{7} + 54025 p^{2} T^{8} - 3015 p^{3} T^{9} + 362 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( ( 1 - 22 T + 565 T^{2} - 7781 T^{3} + 110749 T^{4} - 1067896 T^{5} + 10367459 T^{6} - 1067896 p T^{7} + 110749 p^{2} T^{8} - 7781 p^{3} T^{9} + 565 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( ( 1 - 5 T + 282 T^{2} - 1537 T^{3} + 40228 T^{4} - 200831 T^{5} + 3541613 T^{6} - 200831 p T^{7} + 40228 p^{2} T^{8} - 1537 p^{3} T^{9} + 282 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 73 | \( 1 - 6 T - 264 T^{2} + 676 T^{3} + 41379 T^{4} - 177 T^{5} - 4351292 T^{6} - 7764489 T^{7} + 350907786 T^{8} + 808896697 T^{9} - 24223821882 T^{10} - 30050609763 T^{11} + 1721091747949 T^{12} - 30050609763 p T^{13} - 24223821882 p^{2} T^{14} + 808896697 p^{3} T^{15} + 350907786 p^{4} T^{16} - 7764489 p^{5} T^{17} - 4351292 p^{6} T^{18} - 177 p^{7} T^{19} + 41379 p^{8} T^{20} + 676 p^{9} T^{21} - 264 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( ( 1 + T + 53 T^{2} - 531 T^{3} + 1360 T^{4} - 20890 T^{5} + 639331 T^{6} - 20890 p T^{7} + 1360 p^{2} T^{8} - 531 p^{3} T^{9} + 53 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 + 26 T + 232 T^{2} + 1516 T^{3} + 11499 T^{4} - 94349 T^{5} - 2607716 T^{6} - 23093955 T^{7} - 195776338 T^{8} - 925613923 T^{9} + 9434947014 T^{10} + 206018004805 T^{11} + 2189491903883 T^{12} + 206018004805 p T^{13} + 9434947014 p^{2} T^{14} - 925613923 p^{3} T^{15} - 195776338 p^{4} T^{16} - 23093955 p^{5} T^{17} - 2607716 p^{6} T^{18} - 94349 p^{7} T^{19} + 11499 p^{8} T^{20} + 1516 p^{9} T^{21} + 232 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 2 T - 131 T^{2} + 1310 T^{3} - 5289 T^{4} - 98935 T^{5} + 2008330 T^{6} - 7349421 T^{7} - 31479907 T^{8} + 1608975643 T^{9} - 11307686898 T^{10} - 57565868677 T^{11} + 1401967996337 T^{12} - 57565868677 p T^{13} - 11307686898 p^{2} T^{14} + 1608975643 p^{3} T^{15} - 31479907 p^{4} T^{16} - 7349421 p^{5} T^{17} + 2008330 p^{6} T^{18} - 98935 p^{7} T^{19} - 5289 p^{8} T^{20} + 1310 p^{9} T^{21} - 131 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 6 T - 168 T^{2} + 1492 T^{3} + 5214 T^{4} - 121578 T^{5} + 1943470 T^{6} - 2165256 T^{7} - 213568578 T^{8} + 1521589192 T^{9} - 5900408970 T^{10} - 81665164176 T^{11} + 2384863292767 T^{12} - 81665164176 p T^{13} - 5900408970 p^{2} T^{14} + 1521589192 p^{3} T^{15} - 213568578 p^{4} T^{16} - 2165256 p^{5} T^{17} + 1943470 p^{6} T^{18} - 121578 p^{7} T^{19} + 5214 p^{8} T^{20} + 1492 p^{9} T^{21} - 168 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.44896802652673114692159941694, −3.41499594839722213605594561008, −3.36718704605489796478750716701, −3.29847386187783052726129023250, −2.99833267422153576054359296116, −2.97410953113611712957733774485, −2.97123558910428624665446946636, −2.94077237455305564198697211801, −2.83972434168671100782123792341, −2.62915318203925037001017373340, −2.57893372542462448906829097234, −2.56536800905053654001528566030, −2.39947667107946507941848431195, −2.04366575621503336944638919635, −2.03472799679659722937138719828, −1.86420157251388859054062571516, −1.80204504128574744287980142222, −1.75224994645616744017924673477, −1.74067567616893354586266535086, −1.73448439773393695831013855363, −1.70938659062537190550455283392, −1.39229629365785063434520389982, −1.22516478384214757262745760357, −0.884478094351600216223246375174, −0.71810832815710592829157456733, 0.71810832815710592829157456733, 0.884478094351600216223246375174, 1.22516478384214757262745760357, 1.39229629365785063434520389982, 1.70938659062537190550455283392, 1.73448439773393695831013855363, 1.74067567616893354586266535086, 1.75224994645616744017924673477, 1.80204504128574744287980142222, 1.86420157251388859054062571516, 2.03472799679659722937138719828, 2.04366575621503336944638919635, 2.39947667107946507941848431195, 2.56536800905053654001528566030, 2.57893372542462448906829097234, 2.62915318203925037001017373340, 2.83972434168671100782123792341, 2.94077237455305564198697211801, 2.97123558910428624665446946636, 2.97410953113611712957733774485, 2.99833267422153576054359296116, 3.29847386187783052726129023250, 3.36718704605489796478750716701, 3.41499594839722213605594561008, 3.44896802652673114692159941694
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.