Dirichlet series
| L(s) = 1 | − 24·4-s + 8·5-s + 131·9-s + 62·11-s + 336·16-s − 186·19-s − 192·20-s − 31·25-s − 338·29-s + 652·31-s − 3.14e3·36-s + 396·41-s − 1.48e3·44-s + 1.04e3·45-s + 496·55-s − 1.33e3·59-s + 314·61-s − 3.58e3·64-s + 2.21e3·71-s + 4.46e3·76-s + 1.77e3·79-s + 2.68e3·80-s + 7.85e3·81-s − 6.09e3·89-s − 1.48e3·95-s + 8.12e3·99-s + 744·100-s + ⋯ |
| L(s) = 1 | − 3·4-s + 0.715·5-s + 4.85·9-s + 1.69·11-s + 21/4·16-s − 2.24·19-s − 2.14·20-s − 0.247·25-s − 2.16·29-s + 3.77·31-s − 14.5·36-s + 1.50·41-s − 5.09·44-s + 3.47·45-s + 1.21·55-s − 2.94·59-s + 0.659·61-s − 7·64-s + 3.70·71-s + 6.73·76-s + 2.52·79-s + 3.75·80-s + 10.7·81-s − 7.25·89-s − 1.60·95-s + 8.24·99-s + 0.743·100-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 5^{12} \cdot 7^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.40993\times 10^{17}\) |
| Root analytic conductor: | \(5.37688\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 5^{12} \cdot 7^{24} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(39.44404006\) |
| \(L(\frac12)\) | \(\approx\) | \(39.44404006\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{2} T^{2} )^{6} \) |
| 5 | \( 1 - 8 T + 19 p T^{2} - 2162 T^{3} + 22079 T^{4} - 53046 p T^{5} + 109522 p^{2} T^{6} - 53046 p^{4} T^{7} + 22079 p^{6} T^{8} - 2162 p^{9} T^{9} + 19 p^{13} T^{10} - 8 p^{15} T^{11} + p^{18} T^{12} \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 - 131 T^{2} + 1034 p^{2} T^{4} - 458249 T^{6} + 17513020 T^{8} - 186686789 p T^{10} + 1764074884 p^{2} T^{12} - 186686789 p^{7} T^{14} + 17513020 p^{12} T^{16} - 458249 p^{18} T^{18} + 1034 p^{26} T^{20} - 131 p^{30} T^{22} + p^{36} T^{24} \) |
| 11 | \( ( 1 - 31 T + 4953 T^{2} - 109554 T^{3} + 11770931 T^{4} - 212089231 T^{5} + 18787409958 T^{6} - 212089231 p^{3} T^{7} + 11770931 p^{6} T^{8} - 109554 p^{9} T^{9} + 4953 p^{12} T^{10} - 31 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 13 | \( 1 - 3375 T^{2} + 11747028 T^{4} - 3449732763 T^{6} - 7666714083849 T^{8} + 146214556816892034 T^{10} - 95803398004806680 p^{3} T^{12} + 146214556816892034 p^{6} T^{14} - 7666714083849 p^{12} T^{16} - 3449732763 p^{18} T^{18} + 11747028 p^{24} T^{20} - 3375 p^{30} T^{22} + p^{36} T^{24} \) | |
| 17 | \( 1 - 22662 T^{2} + 291957807 T^{4} - 2787637000470 T^{6} + 21446050462654971 T^{8} - \)\(13\!\cdots\!20\)\( T^{10} + \)\(72\!\cdots\!62\)\( T^{12} - \)\(13\!\cdots\!20\)\( p^{6} T^{14} + 21446050462654971 p^{12} T^{16} - 2787637000470 p^{18} T^{18} + 291957807 p^{24} T^{20} - 22662 p^{30} T^{22} + p^{36} T^{24} \) | |
| 19 | \( ( 1 + 93 T + 30543 T^{2} + 3069034 T^{3} + 431810157 T^{4} + 41104963749 T^{5} + 3693649160078 T^{6} + 41104963749 p^{3} T^{7} + 431810157 p^{6} T^{8} + 3069034 p^{9} T^{9} + 30543 p^{12} T^{10} + 93 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 23 | \( 1 - 73882 T^{2} + 2710550133 T^{4} - 65921886493598 T^{6} + 1200902771331914010 T^{8} - \)\(17\!\cdots\!86\)\( T^{10} + \)\(22\!\cdots\!57\)\( T^{12} - \)\(17\!\cdots\!86\)\( p^{6} T^{14} + 1200902771331914010 p^{12} T^{16} - 65921886493598 p^{18} T^{18} + 2710550133 p^{24} T^{20} - 73882 p^{30} T^{22} + p^{36} T^{24} \) | |
| 29 | \( ( 1 + 169 T + 63805 T^{2} + 9538672 T^{3} + 2096897711 T^{4} + 278097683143 T^{5} + 53594954765494 T^{6} + 278097683143 p^{3} T^{7} + 2096897711 p^{6} T^{8} + 9538672 p^{9} T^{9} + 63805 p^{12} T^{10} + 169 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 31 | \( ( 1 - 326 T + 169807 T^{2} - 37063070 T^{3} + 11075483595 T^{4} - 1823855673672 T^{5} + 410874041723418 T^{6} - 1823855673672 p^{3} T^{7} + 11075483595 p^{6} T^{8} - 37063070 p^{9} T^{9} + 169807 p^{12} T^{10} - 326 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 37 | \( 1 - 323133 T^{2} + 1402015959 p T^{4} - 5496998663247992 T^{6} + \)\(43\!\cdots\!65\)\( T^{8} - \)\(27\!\cdots\!39\)\( T^{10} + \)\(15\!\cdots\!82\)\( T^{12} - \)\(27\!\cdots\!39\)\( p^{6} T^{14} + \)\(43\!\cdots\!65\)\( p^{12} T^{16} - 5496998663247992 p^{18} T^{18} + 1402015959 p^{25} T^{20} - 323133 p^{30} T^{22} + p^{36} T^{24} \) | |
| 41 | \( ( 1 - 198 T + 265890 T^{2} - 43780628 T^{3} + 771850182 p T^{4} - 4393246765554 T^{5} + 2489734823013574 T^{6} - 4393246765554 p^{3} T^{7} + 771850182 p^{7} T^{8} - 43780628 p^{9} T^{9} + 265890 p^{12} T^{10} - 198 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 43 | \( 1 - 468741 T^{2} + 115367946687 T^{4} - 19265293317740372 T^{6} + \)\(24\!\cdots\!89\)\( T^{8} - \)\(25\!\cdots\!23\)\( T^{10} + \)\(21\!\cdots\!94\)\( T^{12} - \)\(25\!\cdots\!23\)\( p^{6} T^{14} + \)\(24\!\cdots\!89\)\( p^{12} T^{16} - 19265293317740372 p^{18} T^{18} + 115367946687 p^{24} T^{20} - 468741 p^{30} T^{22} + p^{36} T^{24} \) | |
| 47 | \( 1 - 610893 T^{2} + 171586449351 T^{4} - 30749106375613080 T^{6} + \)\(42\!\cdots\!77\)\( T^{8} - \)\(51\!\cdots\!95\)\( T^{10} + \)\(56\!\cdots\!62\)\( T^{12} - \)\(51\!\cdots\!95\)\( p^{6} T^{14} + \)\(42\!\cdots\!77\)\( p^{12} T^{16} - 30749106375613080 p^{18} T^{18} + 171586449351 p^{24} T^{20} - 610893 p^{30} T^{22} + p^{36} T^{24} \) | |
| 53 | \( 1 - 681041 T^{2} + 253613903983 T^{4} - 71384165103578360 T^{6} + \)\(16\!\cdots\!09\)\( T^{8} - \)\(31\!\cdots\!15\)\( T^{10} + \)\(50\!\cdots\!94\)\( T^{12} - \)\(31\!\cdots\!15\)\( p^{6} T^{14} + \)\(16\!\cdots\!09\)\( p^{12} T^{16} - 71384165103578360 p^{18} T^{18} + 253613903983 p^{24} T^{20} - 681041 p^{30} T^{22} + p^{36} T^{24} \) | |
| 59 | \( ( 1 + 668 T + 419183 T^{2} + 86779642 T^{3} + 67613720439 T^{4} + 34876007754934 T^{5} + 26417861710422034 T^{6} + 34876007754934 p^{3} T^{7} + 67613720439 p^{6} T^{8} + 86779642 p^{9} T^{9} + 419183 p^{12} T^{10} + 668 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 61 | \( ( 1 - 157 T + 943554 T^{2} - 102791887 T^{3} + 413654135120 T^{4} - 28583021651089 T^{5} + 113959502812162584 T^{6} - 28583021651089 p^{3} T^{7} + 413654135120 p^{6} T^{8} - 102791887 p^{9} T^{9} + 943554 p^{12} T^{10} - 157 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 67 | \( 1 - 2242855 T^{2} + 2403417182822 T^{4} - 1634660374074755909 T^{6} + \)\(80\!\cdots\!28\)\( T^{8} - \)\(30\!\cdots\!63\)\( T^{10} + \)\(10\!\cdots\!48\)\( T^{12} - \)\(30\!\cdots\!63\)\( p^{6} T^{14} + \)\(80\!\cdots\!28\)\( p^{12} T^{16} - 1634660374074755909 p^{18} T^{18} + 2403417182822 p^{24} T^{20} - 2242855 p^{30} T^{22} + p^{36} T^{24} \) | |
| 71 | \( ( 1 - 1108 T + 1271370 T^{2} - 1089287548 T^{3} + 843388393551 T^{4} - 607378211258248 T^{5} + 386679142775954284 T^{6} - 607378211258248 p^{3} T^{7} + 843388393551 p^{6} T^{8} - 1089287548 p^{9} T^{9} + 1271370 p^{12} T^{10} - 1108 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 73 | \( 1 - 2949834 T^{2} + 4318157830959 T^{4} - 4166025249889331630 T^{6} + \)\(29\!\cdots\!15\)\( T^{8} - \)\(16\!\cdots\!40\)\( T^{10} + \)\(70\!\cdots\!30\)\( T^{12} - \)\(16\!\cdots\!40\)\( p^{6} T^{14} + \)\(29\!\cdots\!15\)\( p^{12} T^{16} - 4166025249889331630 p^{18} T^{18} + 4318157830959 p^{24} T^{20} - 2949834 p^{30} T^{22} + p^{36} T^{24} \) | |
| 79 | \( ( 1 - 886 T + 2311007 T^{2} - 1684502530 T^{3} + 2532973731923 T^{4} - 1481255163137980 T^{5} + 1599367231278483338 T^{6} - 1481255163137980 p^{3} T^{7} + 2532973731923 p^{6} T^{8} - 1684502530 p^{9} T^{9} + 2311007 p^{12} T^{10} - 886 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 83 | \( 1 - 1851053 T^{2} + 2192624019487 T^{4} - 1943833747968619300 T^{6} + \)\(14\!\cdots\!81\)\( T^{8} - \)\(10\!\cdots\!75\)\( T^{10} + \)\(63\!\cdots\!82\)\( T^{12} - \)\(10\!\cdots\!75\)\( p^{6} T^{14} + \)\(14\!\cdots\!81\)\( p^{12} T^{16} - 1943833747968619300 p^{18} T^{18} + 2192624019487 p^{24} T^{20} - 1851053 p^{30} T^{22} + p^{36} T^{24} \) | |
| 89 | \( ( 1 + 3047 T + 7083818 T^{2} + 11533239641 T^{3} + 15545102042312 T^{4} + 16815674531849211 T^{5} + 15577186807481084848 T^{6} + 16815674531849211 p^{3} T^{7} + 15545102042312 p^{6} T^{8} + 11533239641 p^{9} T^{9} + 7083818 p^{12} T^{10} + 3047 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 97 | \( 1 - 6512348 T^{2} + 21096769835234 T^{4} - 467896787811259884 p T^{6} + \)\(72\!\cdots\!43\)\( T^{8} - \)\(90\!\cdots\!24\)\( T^{10} + \)\(92\!\cdots\!04\)\( T^{12} - \)\(90\!\cdots\!24\)\( p^{6} T^{14} + \)\(72\!\cdots\!43\)\( p^{12} T^{16} - 467896787811259884 p^{19} T^{18} + 21096769835234 p^{24} T^{20} - 6512348 p^{30} T^{22} + p^{36} T^{24} \) | |
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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
−3.26954330919597258875298188001, −3.07696013318141773430242355706, −2.84037008159359265905638400018, −2.74286210424236871857786727240, −2.70584680432516026505822971871, −2.70453877534792129798822629077, −2.49636112891657591382587486649, −2.17592369423845481583254901411, −2.15788193258017018325941984086, −2.05066307233339095894665573527, −1.94372214355271280225277078300, −1.94007148846914004131253206323, −1.71349828002352603579680542805, −1.38786331496827651559013011527, −1.38585256997986744666990269479, −1.34511835625039183609511327461, −1.30803328998642762290393678325, −1.25876270054082362311691489702, −1.06225241110493665755751759558, −0.823144472130852539112832917364, −0.62330824837801061923975043690, −0.50980479715410204338671273630, −0.46983503328241976188901886569, −0.31734705134313680755478800226, −0.27051402875385728498573809843, 0.27051402875385728498573809843, 0.31734705134313680755478800226, 0.46983503328241976188901886569, 0.50980479715410204338671273630, 0.62330824837801061923975043690, 0.823144472130852539112832917364, 1.06225241110493665755751759558, 1.25876270054082362311691489702, 1.30803328998642762290393678325, 1.34511835625039183609511327461, 1.38585256997986744666990269479, 1.38786331496827651559013011527, 1.71349828002352603579680542805, 1.94007148846914004131253206323, 1.94372214355271280225277078300, 2.05066307233339095894665573527, 2.15788193258017018325941984086, 2.17592369423845481583254901411, 2.49636112891657591382587486649, 2.70453877534792129798822629077, 2.70584680432516026505822971871, 2.74286210424236871857786727240, 2.84037008159359265905638400018, 3.07696013318141773430242355706, 3.26954330919597258875298188001
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.