Dirichlet series
| L(s) = 1 | − 3·2-s − 3·3-s − 2·4-s − 16·5-s + 9·6-s − 7·7-s + 14·8-s − 12·9-s + 48·10-s − 6·11-s + 6·12-s − 15·13-s + 21·14-s + 48·15-s − 3·16-s − 18·17-s + 36·18-s − 6·19-s + 32·20-s + 21·21-s + 18·22-s − 12·23-s − 42·24-s + 105·25-s + 45·26-s + 44·27-s + 14·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s − 4-s − 7.15·5-s + 3.67·6-s − 2.64·7-s + 4.94·8-s − 4·9-s + 15.1·10-s − 1.80·11-s + 1.73·12-s − 4.16·13-s + 5.61·14-s + 12.3·15-s − 3/4·16-s − 4.36·17-s + 8.48·18-s − 1.37·19-s + 7.15·20-s + 4.58·21-s + 3.83·22-s − 2.50·23-s − 8.57·24-s + 21·25-s + 8.82·26-s + 8.46·27-s + 2.64·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(23^{12} \cdot 29^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.20997\times 10^{8}\) |
| Root analytic conductor: | \(2.30781\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 23^{12} \cdot 29^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
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 | 23 | \( ( 1 + T )^{12} \) |
| 29 | \( ( 1 + T )^{12} \) | |
| good | 2 | \( 1 + 3 T + 11 T^{2} + 25 T^{3} + 29 p T^{4} + 55 p T^{5} + 207 T^{6} + 173 p T^{7} + 577 T^{8} + 875 T^{9} + 1349 T^{10} + 1909 T^{11} + 2807 T^{12} + 1909 p T^{13} + 1349 p^{2} T^{14} + 875 p^{3} T^{15} + 577 p^{4} T^{16} + 173 p^{6} T^{17} + 207 p^{6} T^{18} + 55 p^{8} T^{19} + 29 p^{9} T^{20} + 25 p^{9} T^{21} + 11 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) |
| 3 | \( 1 + p T + 7 p T^{2} + 55 T^{3} + 218 T^{4} + 56 p^{2} T^{5} + 497 p T^{6} + 3074 T^{7} + 2524 p T^{8} + 14047 T^{9} + 10148 p T^{10} + 51227 T^{11} + 100313 T^{12} + 51227 p T^{13} + 10148 p^{3} T^{14} + 14047 p^{3} T^{15} + 2524 p^{5} T^{16} + 3074 p^{5} T^{17} + 497 p^{7} T^{18} + 56 p^{9} T^{19} + 218 p^{8} T^{20} + 55 p^{9} T^{21} + 7 p^{11} T^{22} + p^{12} T^{23} + p^{12} T^{24} \) | |
| 5 | \( 1 + 16 T + 151 T^{2} + 1042 T^{3} + 5804 T^{4} + 27347 T^{5} + 112408 T^{6} + 410786 T^{7} + 1353173 T^{8} + 811008 p T^{9} + 11129571 T^{10} + 1123777 p^{2} T^{11} + 65397143 T^{12} + 1123777 p^{3} T^{13} + 11129571 p^{2} T^{14} + 811008 p^{4} T^{15} + 1353173 p^{4} T^{16} + 410786 p^{5} T^{17} + 112408 p^{6} T^{18} + 27347 p^{7} T^{19} + 5804 p^{8} T^{20} + 1042 p^{9} T^{21} + 151 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 + p T + 65 T^{2} + 51 p T^{3} + 2029 T^{4} + 1290 p T^{5} + 39824 T^{6} + 21373 p T^{7} + 550512 T^{8} + 256134 p T^{9} + 812874 p T^{10} + 16257543 T^{11} + 45237078 T^{12} + 16257543 p T^{13} + 812874 p^{3} T^{14} + 256134 p^{4} T^{15} + 550512 p^{4} T^{16} + 21373 p^{6} T^{17} + 39824 p^{6} T^{18} + 1290 p^{8} T^{19} + 2029 p^{8} T^{20} + 51 p^{10} T^{21} + 65 p^{10} T^{22} + p^{12} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 6 T + 69 T^{2} + 283 T^{3} + 2102 T^{4} + 7523 T^{5} + 45973 T^{6} + 153816 T^{7} + 788320 T^{8} + 2463728 T^{9} + 11074500 T^{10} + 32738381 T^{11} + 132642913 T^{12} + 32738381 p T^{13} + 11074500 p^{2} T^{14} + 2463728 p^{3} T^{15} + 788320 p^{4} T^{16} + 153816 p^{5} T^{17} + 45973 p^{6} T^{18} + 7523 p^{7} T^{19} + 2102 p^{8} T^{20} + 283 p^{9} T^{21} + 69 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 15 T + 166 T^{2} + 1273 T^{3} + 8652 T^{4} + 49483 T^{5} + 266267 T^{6} + 1274153 T^{7} + 5845861 T^{8} + 24481580 T^{9} + 100038580 T^{10} + 29201709 p T^{11} + 1417566029 T^{12} + 29201709 p^{2} T^{13} + 100038580 p^{2} T^{14} + 24481580 p^{3} T^{15} + 5845861 p^{4} T^{16} + 1274153 p^{5} T^{17} + 266267 p^{6} T^{18} + 49483 p^{7} T^{19} + 8652 p^{8} T^{20} + 1273 p^{9} T^{21} + 166 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 18 T + 275 T^{2} + 2930 T^{3} + 27660 T^{4} + 217720 T^{5} + 5403 p^{2} T^{6} + 582055 p T^{7} + 58087329 T^{8} + 309237988 T^{9} + 1539835855 T^{10} + 7039278989 T^{11} + 30235172546 T^{12} + 7039278989 p T^{13} + 1539835855 p^{2} T^{14} + 309237988 p^{3} T^{15} + 58087329 p^{4} T^{16} + 582055 p^{6} T^{17} + 5403 p^{8} T^{18} + 217720 p^{7} T^{19} + 27660 p^{8} T^{20} + 2930 p^{9} T^{21} + 275 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 6 T + 109 T^{2} + 357 T^{3} + 4376 T^{4} + 4409 T^{5} + 99783 T^{6} - 124994 T^{7} + 2170382 T^{8} - 4573836 T^{9} + 56249733 T^{10} - 77379748 T^{11} + 1244276232 T^{12} - 77379748 p T^{13} + 56249733 p^{2} T^{14} - 4573836 p^{3} T^{15} + 2170382 p^{4} T^{16} - 124994 p^{5} T^{17} + 99783 p^{6} T^{18} + 4409 p^{7} T^{19} + 4376 p^{8} T^{20} + 357 p^{9} T^{21} + 109 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 16 T + 397 T^{2} - 4753 T^{3} + 67843 T^{4} - 653566 T^{5} + 6860588 T^{6} - 55290793 T^{7} + 466499848 T^{8} - 3212098525 T^{9} + 22711219041 T^{10} - 134878897381 T^{11} + 815898095197 T^{12} - 134878897381 p T^{13} + 22711219041 p^{2} T^{14} - 3212098525 p^{3} T^{15} + 466499848 p^{4} T^{16} - 55290793 p^{5} T^{17} + 6860588 p^{6} T^{18} - 653566 p^{7} T^{19} + 67843 p^{8} T^{20} - 4753 p^{9} T^{21} + 397 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + T + 144 T^{2} + 268 T^{3} + 11671 T^{4} + 16087 T^{5} + 689449 T^{6} + 258592 T^{7} + 31233836 T^{8} - 24273612 T^{9} + 1203162288 T^{10} - 2025452524 T^{11} + 44091152134 T^{12} - 2025452524 p T^{13} + 1203162288 p^{2} T^{14} - 24273612 p^{3} T^{15} + 31233836 p^{4} T^{16} + 258592 p^{5} T^{17} + 689449 p^{6} T^{18} + 16087 p^{7} T^{19} + 11671 p^{8} T^{20} + 268 p^{9} T^{21} + 144 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 3 T + 267 T^{2} - 1253 T^{3} + 38030 T^{4} - 203473 T^{5} + 3795696 T^{6} - 20268147 T^{7} + 284183426 T^{8} - 1439876239 T^{9} + 16521954046 T^{10} - 76833132697 T^{11} + 759993748432 T^{12} - 76833132697 p T^{13} + 16521954046 p^{2} T^{14} - 1439876239 p^{3} T^{15} + 284183426 p^{4} T^{16} - 20268147 p^{5} T^{17} + 3795696 p^{6} T^{18} - 203473 p^{7} T^{19} + 38030 p^{8} T^{20} - 1253 p^{9} T^{21} + 267 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 23 T + 534 T^{2} + 7164 T^{3} + 95402 T^{4} + 896932 T^{5} + 8597959 T^{6} + 60154369 T^{7} + 454291076 T^{8} + 2379928831 T^{9} + 15740252883 T^{10} + 66376399142 T^{11} + 532701233829 T^{12} + 66376399142 p T^{13} + 15740252883 p^{2} T^{14} + 2379928831 p^{3} T^{15} + 454291076 p^{4} T^{16} + 60154369 p^{5} T^{17} + 8597959 p^{6} T^{18} + 896932 p^{7} T^{19} + 95402 p^{8} T^{20} + 7164 p^{9} T^{21} + 534 p^{10} T^{22} + 23 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 35 T + 786 T^{2} + 13088 T^{3} + 181182 T^{4} + 2157152 T^{5} + 23003847 T^{6} + 223443780 T^{7} + 2017433324 T^{8} + 17029593972 T^{9} + 135477644983 T^{10} + 1014476657609 T^{11} + 7169245407477 T^{12} + 1014476657609 p T^{13} + 135477644983 p^{2} T^{14} + 17029593972 p^{3} T^{15} + 2017433324 p^{4} T^{16} + 223443780 p^{5} T^{17} + 23003847 p^{6} T^{18} + 2157152 p^{7} T^{19} + 181182 p^{8} T^{20} + 13088 p^{9} T^{21} + 786 p^{10} T^{22} + 35 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 45 T + 1319 T^{2} + 27726 T^{3} + 472754 T^{4} + 6715967 T^{5} + 82858250 T^{6} + 900211755 T^{7} + 8822903856 T^{8} + 78903314982 T^{9} + 656800907646 T^{10} + 5138994786863 T^{11} + 38323412840667 T^{12} + 5138994786863 p T^{13} + 656800907646 p^{2} T^{14} + 78903314982 p^{3} T^{15} + 8822903856 p^{4} T^{16} + 900211755 p^{5} T^{17} + 82858250 p^{6} T^{18} + 6715967 p^{7} T^{19} + 472754 p^{8} T^{20} + 27726 p^{9} T^{21} + 1319 p^{10} T^{22} + 45 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 11 T + 322 T^{2} + 3387 T^{3} + 58100 T^{4} + 586022 T^{5} + 7471787 T^{6} + 70370122 T^{7} + 744998638 T^{8} + 6427574750 T^{9} + 59555314996 T^{10} + 467466370186 T^{11} + 3886847698992 T^{12} + 467466370186 p T^{13} + 59555314996 p^{2} T^{14} + 6427574750 p^{3} T^{15} + 744998638 p^{4} T^{16} + 70370122 p^{5} T^{17} + 7471787 p^{6} T^{18} + 586022 p^{7} T^{19} + 58100 p^{8} T^{20} + 3387 p^{9} T^{21} + 322 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 4 T + 444 T^{2} - 1466 T^{3} + 93276 T^{4} - 240684 T^{5} + 12441225 T^{6} - 22951404 T^{7} + 1203204535 T^{8} - 1421375134 T^{9} + 92409857416 T^{10} - 69967907024 T^{11} + 6031571155150 T^{12} - 69967907024 p T^{13} + 92409857416 p^{2} T^{14} - 1421375134 p^{3} T^{15} + 1203204535 p^{4} T^{16} - 22951404 p^{5} T^{17} + 12441225 p^{6} T^{18} - 240684 p^{7} T^{19} + 93276 p^{8} T^{20} - 1466 p^{9} T^{21} + 444 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 19 T + 582 T^{2} + 8498 T^{3} + 157935 T^{4} + 1947356 T^{5} + 27739678 T^{6} + 298026009 T^{7} + 3521843485 T^{8} + 33428296182 T^{9} + 340561826459 T^{10} + 2872813898074 T^{11} + 25727396379516 T^{12} + 2872813898074 p T^{13} + 340561826459 p^{2} T^{14} + 33428296182 p^{3} T^{15} + 3521843485 p^{4} T^{16} + 298026009 p^{5} T^{17} + 27739678 p^{6} T^{18} + 1947356 p^{7} T^{19} + 157935 p^{8} T^{20} + 8498 p^{9} T^{21} + 582 p^{10} T^{22} + 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 19 T + 557 T^{2} - 8669 T^{3} + 147428 T^{4} - 1886187 T^{5} + 24513358 T^{6} - 263684591 T^{7} + 2890644180 T^{8} - 27203252247 T^{9} + 264256344608 T^{10} - 2267894906725 T^{11} + 20188824935908 T^{12} - 2267894906725 p T^{13} + 264256344608 p^{2} T^{14} - 27203252247 p^{3} T^{15} + 2890644180 p^{4} T^{16} - 263684591 p^{5} T^{17} + 24513358 p^{6} T^{18} - 1886187 p^{7} T^{19} + 147428 p^{8} T^{20} - 8669 p^{9} T^{21} + 557 p^{10} T^{22} - 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 10 T + 470 T^{2} - 5607 T^{3} + 123348 T^{4} - 1452955 T^{5} + 22746543 T^{6} - 243632885 T^{7} + 3115488575 T^{8} - 29875682425 T^{9} + 326956400646 T^{10} - 2804767203394 T^{11} + 26870758635114 T^{12} - 2804767203394 p T^{13} + 326956400646 p^{2} T^{14} - 29875682425 p^{3} T^{15} + 3115488575 p^{4} T^{16} - 243632885 p^{5} T^{17} + 22746543 p^{6} T^{18} - 1452955 p^{7} T^{19} + 123348 p^{8} T^{20} - 5607 p^{9} T^{21} + 470 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 17 T + 726 T^{2} - 9471 T^{3} + 233843 T^{4} - 2476541 T^{5} + 46166955 T^{6} - 412420874 T^{7} + 6460192964 T^{8} - 50180892217 T^{9} + 696914226932 T^{10} - 4825821191556 T^{11} + 60765805192315 T^{12} - 4825821191556 p T^{13} + 696914226932 p^{2} T^{14} - 50180892217 p^{3} T^{15} + 6460192964 p^{4} T^{16} - 412420874 p^{5} T^{17} + 46166955 p^{6} T^{18} - 2476541 p^{7} T^{19} + 233843 p^{8} T^{20} - 9471 p^{9} T^{21} + 726 p^{10} T^{22} - 17 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 12 T + 358 T^{2} + 6137 T^{3} + 78325 T^{4} + 1167568 T^{5} + 13032043 T^{6} + 133237621 T^{7} + 1396486127 T^{8} + 11948659334 T^{9} + 105414867948 T^{10} + 932526228440 T^{11} + 7908158278756 T^{12} + 932526228440 p T^{13} + 105414867948 p^{2} T^{14} + 11948659334 p^{3} T^{15} + 1396486127 p^{4} T^{16} + 133237621 p^{5} T^{17} + 13032043 p^{6} T^{18} + 1167568 p^{7} T^{19} + 78325 p^{8} T^{20} + 6137 p^{9} T^{21} + 358 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 20 T + 684 T^{2} + 118 p T^{3} + 216198 T^{4} + 2804069 T^{5} + 44593072 T^{6} + 509986178 T^{7} + 6856781124 T^{8} + 70502587428 T^{9} + 834633343019 T^{10} + 7761428330603 T^{11} + 82316339528600 T^{12} + 7761428330603 p T^{13} + 834633343019 p^{2} T^{14} + 70502587428 p^{3} T^{15} + 6856781124 p^{4} T^{16} + 509986178 p^{5} T^{17} + 44593072 p^{6} T^{18} + 2804069 p^{7} T^{19} + 216198 p^{8} T^{20} + 118 p^{10} T^{21} + 684 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 12 T + 796 T^{2} + 7580 T^{3} + 297199 T^{4} + 2298359 T^{5} + 70288656 T^{6} + 447238351 T^{7} + 11967689156 T^{8} + 63828920543 T^{9} + 1579738479959 T^{10} + 7308273286367 T^{11} + 168939726182442 T^{12} + 7308273286367 p T^{13} + 1579738479959 p^{2} T^{14} + 63828920543 p^{3} T^{15} + 11967689156 p^{4} T^{16} + 447238351 p^{5} T^{17} + 70288656 p^{6} T^{18} + 2298359 p^{7} T^{19} + 297199 p^{8} T^{20} + 7580 p^{9} T^{21} + 796 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−4.08350492745012857982159777156, −3.85340795580896505324188564759, −3.76917668349945583766218472846, −3.74339274849255335300092625221, −3.57378937888852676414161776274, −3.46680428167158991295790246357, −3.40321367935097835868040985099, −3.36939497005853789646029908319, −3.33468071497689821343157686787, −3.33193960836014992786800795765, −3.12163582341770630101730797076, −3.09372217530364009969402072205, −2.72160749794103743184553795527, −2.64806495271161451677521196668, −2.64227707730943926991536161550, −2.61920845897709971750264121001, −2.58809866124236602303542578108, −2.28873148282654708331106701135, −2.19552189638085621948568313986, −2.05427596932475585611029400552, −2.00162742897713651363276151262, −1.59468185351796211785457369986, −1.46702424631618003327784958468, −1.41001320726794993694899411484, −1.01067586470011474398636321036, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.01067586470011474398636321036, 1.41001320726794993694899411484, 1.46702424631618003327784958468, 1.59468185351796211785457369986, 2.00162742897713651363276151262, 2.05427596932475585611029400552, 2.19552189638085621948568313986, 2.28873148282654708331106701135, 2.58809866124236602303542578108, 2.61920845897709971750264121001, 2.64227707730943926991536161550, 2.64806495271161451677521196668, 2.72160749794103743184553795527, 3.09372217530364009969402072205, 3.12163582341770630101730797076, 3.33193960836014992786800795765, 3.33468071497689821343157686787, 3.36939497005853789646029908319, 3.40321367935097835868040985099, 3.46680428167158991295790246357, 3.57378937888852676414161776274, 3.74339274849255335300092625221, 3.76917668349945583766218472846, 3.85340795580896505324188564759, 4.08350492745012857982159777156
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.