Dirichlet series
| L(s) = 1 | + 2-s − 3-s − 4-s + 3·5-s − 6-s − 7-s − 5·8-s + 4·9-s + 3·10-s + 12-s + 6·13-s − 14-s − 3·15-s − 2·16-s + 4·17-s + 4·18-s + 4·19-s − 3·20-s + 21-s − 24·23-s + 5·24-s + 3·25-s + 6·26-s − 7·27-s + 28-s + 2·29-s − 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.377·7-s − 1.76·8-s + 4/3·9-s + 0.948·10-s + 0.288·12-s + 1.66·13-s − 0.267·14-s − 0.774·15-s − 1/2·16-s + 0.970·17-s + 0.942·18-s + 0.917·19-s − 0.670·20-s + 0.218·21-s − 5.00·23-s + 1.02·24-s + 3/5·25-s + 1.17·26-s − 1.34·27-s + 0.188·28-s + 0.371·29-s − 0.547·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 11^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 11^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(5^{12} \cdot 11^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.61580\times 10^{8}\) |
| Root analytic conductor: | \(2.19794\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 5^{12} \cdot 11^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(10.58373662\) |
| \(L(\frac12)\) | \(\approx\) | \(10.58373662\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{3} \) |
| 11 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} + p T^{3} - 3 T^{4} + 7 T^{6} - 9 p T^{7} - 5 T^{8} + 9 p T^{9} - 13 p T^{10} - 45 T^{11} + 115 T^{12} - 45 p T^{13} - 13 p^{3} T^{14} + 9 p^{4} T^{15} - 5 p^{4} T^{16} - 9 p^{6} T^{17} + 7 p^{6} T^{18} - 3 p^{8} T^{20} + p^{10} T^{21} + p^{11} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) |
| 3 | \( 1 + T - p T^{2} + 7 T^{4} + 14 p T^{5} + 32 T^{6} - 47 p T^{7} + 22 T^{8} + 41 p^{2} T^{9} + 97 p^{2} T^{10} + 397 T^{11} - 2909 T^{12} + 397 p T^{13} + 97 p^{4} T^{14} + 41 p^{5} T^{15} + 22 p^{4} T^{16} - 47 p^{6} T^{17} + 32 p^{6} T^{18} + 14 p^{8} T^{19} + 7 p^{8} T^{20} - p^{11} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 + T - T^{2} - 26 T^{3} - 61 T^{4} - 36 p T^{5} + 188 T^{6} + 2067 T^{7} + 7410 T^{8} + 8409 T^{9} + 7283 T^{10} - 113255 T^{11} - 388169 T^{12} - 113255 p T^{13} + 7283 p^{2} T^{14} + 8409 p^{3} T^{15} + 7410 p^{4} T^{16} + 2067 p^{5} T^{17} + 188 p^{6} T^{18} - 36 p^{8} T^{19} - 61 p^{8} T^{20} - 26 p^{9} T^{21} - p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 6 T - 7 T^{2} + 148 T^{3} - 18 p T^{4} - 1770 T^{5} + 5986 T^{6} + 13310 T^{7} - 72105 T^{8} - 148022 T^{9} + 1081434 T^{10} + 1214898 T^{11} - 17632593 T^{12} + 1214898 p T^{13} + 1081434 p^{2} T^{14} - 148022 p^{3} T^{15} - 72105 p^{4} T^{16} + 13310 p^{5} T^{17} + 5986 p^{6} T^{18} - 1770 p^{7} T^{19} - 18 p^{9} T^{20} + 148 p^{9} T^{21} - 7 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 4 T - 19 T^{2} + 128 T^{3} - 90 T^{4} - 1380 T^{5} + 8422 T^{6} - 12012 T^{7} - 126353 T^{8} + 480972 T^{9} + 243358 T^{10} - 2497140 T^{11} + 6834115 T^{12} - 2497140 p T^{13} + 243358 p^{2} T^{14} + 480972 p^{3} T^{15} - 126353 p^{4} T^{16} - 12012 p^{5} T^{17} + 8422 p^{6} T^{18} - 1380 p^{7} T^{19} - 90 p^{8} T^{20} + 128 p^{9} T^{21} - 19 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 4 T - 29 T^{2} + 180 T^{3} + 194 T^{4} - 3112 T^{5} + 6550 T^{6} + 6748 T^{7} - 187629 T^{8} + 833060 T^{9} + 246214 T^{10} - 10590124 T^{11} + 40398139 T^{12} - 10590124 p T^{13} + 246214 p^{2} T^{14} + 833060 p^{3} T^{15} - 187629 p^{4} T^{16} + 6748 p^{5} T^{17} + 6550 p^{6} T^{18} - 3112 p^{7} T^{19} + 194 p^{8} T^{20} + 180 p^{9} T^{21} - 29 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( ( 1 + 6 T + 73 T^{2} + 264 T^{3} + 73 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{4} \) | |
| 29 | \( 1 - 2 T - 55 T^{2} + 184 T^{3} + 1254 T^{4} + 2478 T^{5} - 13958 T^{6} - 346998 T^{7} + 294799 T^{8} + 9660774 T^{9} + 904922 p T^{10} - 78687306 T^{11} - 1396349777 T^{12} - 78687306 p T^{13} + 904922 p^{3} T^{14} + 9660774 p^{3} T^{15} + 294799 p^{4} T^{16} - 346998 p^{5} T^{17} - 13958 p^{6} T^{18} + 2478 p^{7} T^{19} + 1254 p^{8} T^{20} + 184 p^{9} T^{21} - 55 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 14 T + 55 T^{2} - 300 T^{3} - 3670 T^{4} - 11974 T^{5} - 3206 T^{6} - 40034 T^{7} - 566061 T^{8} + 6721190 T^{9} + 74111626 T^{10} + 66728354 T^{11} - 1408441409 T^{12} + 66728354 p T^{13} + 74111626 p^{2} T^{14} + 6721190 p^{3} T^{15} - 566061 p^{4} T^{16} - 40034 p^{5} T^{17} - 3206 p^{6} T^{18} - 11974 p^{7} T^{19} - 3670 p^{8} T^{20} - 300 p^{9} T^{21} + 55 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 4 T - 71 T^{2} - 320 T^{3} + 2926 T^{4} + 33188 T^{5} - 21850 T^{6} - 1623172 T^{7} - 1811833 T^{8} + 52550996 T^{9} + 311796798 T^{10} - 465773052 T^{11} - 14542934277 T^{12} - 465773052 p T^{13} + 311796798 p^{2} T^{14} + 52550996 p^{3} T^{15} - 1811833 p^{4} T^{16} - 1623172 p^{5} T^{17} - 21850 p^{6} T^{18} + 33188 p^{7} T^{19} + 2926 p^{8} T^{20} - 320 p^{9} T^{21} - 71 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 9 T + 3 T^{2} + 234 T^{3} - 1569 T^{4} - 4932 T^{5} + 121222 T^{6} - 187803 T^{7} - 4319364 T^{8} + 23446431 T^{9} - 87449985 T^{10} - 30387501 T^{11} + 5054566465 T^{12} - 30387501 p T^{13} - 87449985 p^{2} T^{14} + 23446431 p^{3} T^{15} - 4319364 p^{4} T^{16} - 187803 p^{5} T^{17} + 121222 p^{6} T^{18} - 4932 p^{7} T^{19} - 1569 p^{8} T^{20} + 234 p^{9} T^{21} + 3 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( ( 1 + 7 T + 126 T^{2} + 539 T^{3} + 126 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{4} \) | |
| 47 | \( 1 - 15 T + 113 T^{2} - 624 T^{3} + 575 T^{4} - 36594 T^{5} + 662896 T^{6} - 5097261 T^{7} + 30417750 T^{8} - 50712939 T^{9} + 618298777 T^{10} - 13452820431 T^{11} + 103381879119 T^{12} - 13452820431 p T^{13} + 618298777 p^{2} T^{14} - 50712939 p^{3} T^{15} + 30417750 p^{4} T^{16} - 5097261 p^{5} T^{17} + 662896 p^{6} T^{18} - 36594 p^{7} T^{19} + 575 p^{8} T^{20} - 624 p^{9} T^{21} + 113 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 6 T - 127 T^{2} + 1116 T^{3} + 9110 T^{4} - 64674 T^{5} - 743870 T^{6} + 2713038 T^{7} + 66726183 T^{8} - 139375398 T^{9} - 4084183334 T^{10} + 4313872482 T^{11} + 215380993503 T^{12} + 4313872482 p T^{13} - 4084183334 p^{2} T^{14} - 139375398 p^{3} T^{15} + 66726183 p^{4} T^{16} + 2713038 p^{5} T^{17} - 743870 p^{6} T^{18} - 64674 p^{7} T^{19} + 9110 p^{8} T^{20} + 1116 p^{9} T^{21} - 127 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 10 T - 25 T^{2} - 668 T^{3} - 2610 T^{4} + 40110 T^{5} + 498670 T^{6} - 196110 T^{7} - 29793185 T^{8} - 107996838 T^{9} + 52866790 T^{10} - 192395010 T^{11} + 12096136279 T^{12} - 192395010 p T^{13} + 52866790 p^{2} T^{14} - 107996838 p^{3} T^{15} - 29793185 p^{4} T^{16} - 196110 p^{5} T^{17} + 498670 p^{6} T^{18} + 40110 p^{7} T^{19} - 2610 p^{8} T^{20} - 668 p^{9} T^{21} - 25 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 3 T - 13 T^{2} + 658 T^{3} - 4689 T^{4} + 61428 T^{5} + 106702 T^{6} - 4321513 T^{7} + 46484916 T^{8} - 164821823 T^{9} + 485278491 T^{10} + 18541770333 T^{11} - 195808940235 T^{12} + 18541770333 p T^{13} + 485278491 p^{2} T^{14} - 164821823 p^{3} T^{15} + 46484916 p^{4} T^{16} - 4321513 p^{5} T^{17} + 106702 p^{6} T^{18} + 61428 p^{7} T^{19} - 4689 p^{8} T^{20} + 658 p^{9} T^{21} - 13 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( ( 1 - 19 T + 296 T^{2} - 2605 T^{3} + 296 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} )^{4} \) | |
| 71 | \( 1 + 6 T - 181 T^{2} - 1548 T^{3} + 19622 T^{4} + 124506 T^{5} - 2199026 T^{6} - 105654 p T^{7} + 252410007 T^{8} + 524358678 T^{9} - 20759568698 T^{10} - 21818053158 T^{11} + 1485665622111 T^{12} - 21818053158 p T^{13} - 20759568698 p^{2} T^{14} + 524358678 p^{3} T^{15} + 252410007 p^{4} T^{16} - 105654 p^{6} T^{17} - 2199026 p^{6} T^{18} + 124506 p^{7} T^{19} + 19622 p^{8} T^{20} - 1548 p^{9} T^{21} - 181 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 12 T - 91 T^{2} - 2192 T^{3} - 1434 T^{4} + 159708 T^{5} + 422710 T^{6} - 7809052 T^{7} + 1479087 T^{8} + 601170556 T^{9} + 1139186574 T^{10} - 26194193028 T^{11} - 237214175757 T^{12} - 26194193028 p T^{13} + 1139186574 p^{2} T^{14} + 601170556 p^{3} T^{15} + 1479087 p^{4} T^{16} - 7809052 p^{5} T^{17} + 422710 p^{6} T^{18} + 159708 p^{7} T^{19} - 1434 p^{8} T^{20} - 2192 p^{9} T^{21} - 91 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 2 T - 157 T^{2} + 616 T^{3} + 11366 T^{4} + 66510 T^{5} - 446458 T^{6} - 14931666 T^{7} + 25566927 T^{8} + 1070270406 T^{9} + 5554942910 T^{10} - 21238235006 T^{11} - 791962230257 T^{12} - 21238235006 p T^{13} + 5554942910 p^{2} T^{14} + 1070270406 p^{3} T^{15} + 25566927 p^{4} T^{16} - 14931666 p^{5} T^{17} - 446458 p^{6} T^{18} + 66510 p^{7} T^{19} + 11366 p^{8} T^{20} + 616 p^{9} T^{21} - 157 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 18 T + 39 T^{2} + 1764 T^{3} - 18570 T^{4} - 70830 T^{5} + 2605798 T^{6} - 13518666 T^{7} - 112391193 T^{8} + 1903583934 T^{9} - 7909728498 T^{10} - 63576616230 T^{11} + 1172497984615 T^{12} - 63576616230 p T^{13} - 7909728498 p^{2} T^{14} + 1903583934 p^{3} T^{15} - 112391193 p^{4} T^{16} - 13518666 p^{5} T^{17} + 2605798 p^{6} T^{18} - 70830 p^{7} T^{19} - 18570 p^{8} T^{20} + 1764 p^{9} T^{21} + 39 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( ( 1 - 11 T + 110 T^{2} - 239 T^{3} + 110 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} )^{4} \) | |
| 97 | \( 1 - 2 T - 59 T^{2} - 1076 T^{3} + 2398 T^{4} - 184942 T^{5} + 1124642 T^{6} + 15747746 T^{7} + 201217199 T^{8} - 811978690 T^{9} + 17235000186 T^{10} - 197528464578 T^{11} - 2171345987145 T^{12} - 197528464578 p T^{13} + 17235000186 p^{2} T^{14} - 811978690 p^{3} T^{15} + 201217199 p^{4} T^{16} + 15747746 p^{5} T^{17} + 1124642 p^{6} T^{18} - 184942 p^{7} T^{19} + 2398 p^{8} T^{20} - 1076 p^{9} T^{21} - 59 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.59662415216421415689688900884, −3.53612321860679251443681259695, −3.43133531859961116046659725146, −3.23642956330706207767427032728, −3.18604077487082636048936129588, −3.06328254784705293438106030270, −2.80660902112801163439018899173, −2.80263923412488091099660853651, −2.71271372189689586299819172247, −2.52244999329858985739773089420, −2.31004450307599720987433581017, −2.29321481546453371944435435474, −2.02763020474174117394233205100, −1.97953740234973938547466762769, −1.91993562946127223771998850631, −1.91023291410505916826976043740, −1.86247716612169825439855667381, −1.66370848278791963564489097589, −1.44226567775975255080256234221, −1.12969723790870389689583605918, −1.11801394251362155253791475136, −0.75629095680058615664235929600, −0.71649961989314377984890090339, −0.45859017431773310325624719116, −0.38858492215646682007864731278, 0.38858492215646682007864731278, 0.45859017431773310325624719116, 0.71649961989314377984890090339, 0.75629095680058615664235929600, 1.11801394251362155253791475136, 1.12969723790870389689583605918, 1.44226567775975255080256234221, 1.66370848278791963564489097589, 1.86247716612169825439855667381, 1.91023291410505916826976043740, 1.91993562946127223771998850631, 1.97953740234973938547466762769, 2.02763020474174117394233205100, 2.29321481546453371944435435474, 2.31004450307599720987433581017, 2.52244999329858985739773089420, 2.71271372189689586299819172247, 2.80263923412488091099660853651, 2.80660902112801163439018899173, 3.06328254784705293438106030270, 3.18604077487082636048936129588, 3.23642956330706207767427032728, 3.43133531859961116046659725146, 3.53612321860679251443681259695, 3.59662415216421415689688900884
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.