Dirichlet series
| L(s) = 1 | − 2·2-s + 2·5-s − 25·9-s − 4·10-s + 24·13-s + 8·16-s + 2·17-s + 50·18-s + 77·25-s − 48·26-s + 72·29-s − 32·32-s − 4·34-s + 86·37-s − 8·41-s − 50·45-s − 154·50-s − 74·53-s − 144·58-s − 86·61-s + 16·64-s + 48·65-s + 234·73-s − 172·74-s + 16·80-s + 358·81-s + 16·82-s + ⋯ |
| L(s) = 1 | − 2-s + 2/5·5-s − 2.77·9-s − 2/5·10-s + 1.84·13-s + 1/2·16-s + 2/17·17-s + 25/9·18-s + 3.07·25-s − 1.84·26-s + 2.48·29-s − 32-s − 0.117·34-s + 2.32·37-s − 0.195·41-s − 1.11·45-s − 3.07·50-s − 1.39·53-s − 2.48·58-s − 1.40·61-s + 1/4·64-s + 0.738·65-s + 3.20·73-s − 2.32·74-s + 1/5·80-s + 4.41·81-s + 8/41·82-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 7^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.38385\times 10^{8}\) |
| Root analytic conductor: | \(2.31097\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 7^{24} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.1074455097\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1074455097\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{3} T^{4} + p^{5} T^{5} + 5 p^{4} T^{6} + p^{7} T^{7} + p^{7} T^{8} + p^{9} T^{9} + p^{10} T^{10} + p^{11} T^{11} + p^{12} T^{12} \) |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + 25 T^{2} + 89 p T^{4} + 644 p T^{6} + 11441 T^{8} - 29653 T^{10} - 1105946 T^{12} - 29653 p^{4} T^{14} + 11441 p^{8} T^{16} + 644 p^{13} T^{18} + 89 p^{17} T^{20} + 25 p^{20} T^{22} + p^{24} T^{24} \) |
| 5 | \( ( 1 - T - 37 T^{2} - 92 T^{3} + 521 T^{4} + 2301 T^{5} - 4746 T^{6} + 2301 p^{2} T^{7} + 521 p^{4} T^{8} - 92 p^{6} T^{9} - 37 p^{8} T^{10} - p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 11 | \( 1 + 625 T^{2} + 219387 T^{4} + 53661164 T^{6} + 10139548961 T^{8} + 1568956103859 T^{10} + 205124285853862 T^{12} + 1568956103859 p^{4} T^{14} + 10139548961 p^{8} T^{16} + 53661164 p^{12} T^{18} + 219387 p^{16} T^{20} + 625 p^{20} T^{22} + p^{24} T^{24} \) | |
| 13 | \( ( 1 - 6 T + 431 T^{2} - 1748 T^{3} + 431 p^{2} T^{4} - 6 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 17 | \( ( 1 - T - 21 p T^{2} + 9268 T^{3} + 19889 T^{4} - 1599651 T^{5} + 34317190 T^{6} - 1599651 p^{2} T^{7} + 19889 p^{4} T^{8} + 9268 p^{6} T^{9} - 21 p^{9} T^{10} - p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 19 | \( 1 + 1361 T^{2} + 926139 T^{4} + 455129644 T^{6} + 189034671233 T^{8} + 67541974765395 T^{10} + 23345482892884198 T^{12} + 67541974765395 p^{4} T^{14} + 189034671233 p^{8} T^{16} + 455129644 p^{12} T^{18} + 926139 p^{16} T^{20} + 1361 p^{20} T^{22} + p^{24} T^{24} \) | |
| 23 | \( 1 + 1529 T^{2} + 1564875 T^{4} + 666601324 T^{6} - 12750232255 T^{8} - 342590421568629 T^{10} - 240073619368345850 T^{12} - 342590421568629 p^{4} T^{14} - 12750232255 p^{8} T^{16} + 666601324 p^{12} T^{18} + 1564875 p^{16} T^{20} + 1529 p^{20} T^{22} + p^{24} T^{24} \) | |
| 29 | \( ( 1 - 18 T + 1791 T^{2} - 1196 p T^{3} + 1791 p^{2} T^{4} - 18 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 31 | \( 1 + 1537 T^{2} + 64443 T^{4} - 657806548 T^{6} - 61517921455 T^{8} - 398864870588157 T^{10} - 927701155689421178 T^{12} - 398864870588157 p^{4} T^{14} - 61517921455 p^{8} T^{16} - 657806548 p^{12} T^{18} + 64443 p^{16} T^{20} + 1537 p^{20} T^{22} + p^{24} T^{24} \) | |
| 37 | \( ( 1 - 43 T - 997 T^{2} + 25300 T^{3} + 1028465 T^{4} + 44336247 T^{5} - 3970184154 T^{6} + 44336247 p^{2} T^{7} + 1028465 p^{4} T^{8} + 25300 p^{6} T^{9} - 997 p^{8} T^{10} - 43 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 41 | \( ( 1 + 2 T + 2711 T^{2} - 37572 T^{3} + 2711 p^{2} T^{4} + 2 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
| 43 | \( ( 1 - 4118 T^{2} + 5077855 T^{4} - 2161925300 T^{6} + 5077855 p^{4} T^{8} - 4118 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 47 | \( 1 + 8513 T^{2} + 34636347 T^{4} + 114973411756 T^{6} + 365269331235857 T^{8} + 977402417239620291 T^{10} + \)\(22\!\cdots\!94\)\( T^{12} + 977402417239620291 p^{4} T^{14} + 365269331235857 p^{8} T^{16} + 114973411756 p^{12} T^{18} + 34636347 p^{16} T^{20} + 8513 p^{20} T^{22} + p^{24} T^{24} \) | |
| 53 | \( ( 1 + 37 T - 6101 T^{2} - 98492 T^{3} + 27882353 T^{4} + 139373047 T^{5} - 88083945850 T^{6} + 139373047 p^{2} T^{7} + 27882353 p^{4} T^{8} - 98492 p^{6} T^{9} - 6101 p^{8} T^{10} + 37 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 59 | \( 1 + 8857 T^{2} + 19615883 T^{4} + 62334311372 T^{6} + 716687290054865 T^{8} + 2166290925668370603 T^{10} + \)\(27\!\cdots\!42\)\( T^{12} + 2166290925668370603 p^{4} T^{14} + 716687290054865 p^{8} T^{16} + 62334311372 p^{12} T^{18} + 19615883 p^{16} T^{20} + 8857 p^{20} T^{22} + p^{24} T^{24} \) | |
| 61 | \( ( 1 + 43 T - 7509 T^{2} - 86404 T^{3} + 42241889 T^{4} - 112159143 T^{5} - 188685362330 T^{6} - 112159143 p^{2} T^{7} + 42241889 p^{4} T^{8} - 86404 p^{6} T^{9} - 7509 p^{8} T^{10} + 43 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 67 | \( 1 + 22377 T^{2} + 274481131 T^{4} + 2446555048844 T^{6} + 17357874347837969 T^{8} + \)\(10\!\cdots\!23\)\( T^{10} + \)\(49\!\cdots\!50\)\( T^{12} + \)\(10\!\cdots\!23\)\( p^{4} T^{14} + 17357874347837969 p^{8} T^{16} + 2446555048844 p^{12} T^{18} + 274481131 p^{16} T^{20} + 22377 p^{20} T^{22} + p^{24} T^{24} \) | |
| 71 | \( ( 1 - 18598 T^{2} + 175808431 T^{4} - 1076604135508 T^{6} + 175808431 p^{4} T^{8} - 18598 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 73 | \( ( 1 - 117 T - 6509 T^{2} + 220196 T^{3} + 149197385 T^{4} - 4225343807 T^{5} - 591649860826 T^{6} - 4225343807 p^{2} T^{7} + 149197385 p^{4} T^{8} + 220196 p^{6} T^{9} - 6509 p^{8} T^{10} - 117 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 79 | \( 1 + 19737 T^{2} + 240943563 T^{4} + 1326301878892 T^{6} + 439440672698145 T^{8} - 79155928359774252117 T^{10} - \)\(68\!\cdots\!18\)\( T^{12} - 79155928359774252117 p^{4} T^{14} + 439440672698145 p^{8} T^{16} + 1326301878892 p^{12} T^{18} + 240943563 p^{16} T^{20} + 19737 p^{20} T^{22} + p^{24} T^{24} \) | |
| 83 | \( ( 1 - 38390 T^{2} + 633108223 T^{4} - 5732720590964 T^{6} + 633108223 p^{4} T^{8} - 38390 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 89 | \( ( 1 + 3 T - 22925 T^{2} - 11788 T^{3} + 344187257 T^{4} + 22390969 T^{5} - 3171173759098 T^{6} + 22390969 p^{2} T^{7} + 344187257 p^{4} T^{8} - 11788 p^{6} T^{9} - 22925 p^{8} T^{10} + 3 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 97 | \( ( 1 + 186 T + 27543 T^{2} + 2505868 T^{3} + 27543 p^{2} T^{4} + 186 p^{4} T^{5} + p^{6} T^{6} )^{4} \) | |
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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
−4.10273681709212104939588567455, −3.94909486351014969279362036454, −3.86576447635327356583073043463, −3.67197907232839940691028454805, −3.66613133725641140758309240464, −3.44760688415722532903081516947, −3.17190294059568613294584679340, −3.12296956085939311657041117756, −3.01033888818585607384734186898, −2.94701717974816393503617354990, −2.76262848271180474615018930263, −2.76243405198886621006867325508, −2.53135107650342755875308319608, −2.51534598866109750890784813714, −2.46961307206156994726481912273, −2.16804749866837678684464908562, −1.76886528970525671322371159016, −1.57646254864855056903075001392, −1.49517887167533391070996621454, −1.21048367363370608896198374422, −1.19966812775417535970330584217, −0.969748399666539646741714370101, −0.74314815763159648118783631158, −0.40555283804303528471795465422, −0.05262164928844833694818795486, 0.05262164928844833694818795486, 0.40555283804303528471795465422, 0.74314815763159648118783631158, 0.969748399666539646741714370101, 1.19966812775417535970330584217, 1.21048367363370608896198374422, 1.49517887167533391070996621454, 1.57646254864855056903075001392, 1.76886528970525671322371159016, 2.16804749866837678684464908562, 2.46961307206156994726481912273, 2.51534598866109750890784813714, 2.53135107650342755875308319608, 2.76243405198886621006867325508, 2.76262848271180474615018930263, 2.94701717974816393503617354990, 3.01033888818585607384734186898, 3.12296956085939311657041117756, 3.17190294059568613294584679340, 3.44760688415722532903081516947, 3.66613133725641140758309240464, 3.67197907232839940691028454805, 3.86576447635327356583073043463, 3.94909486351014969279362036454, 4.10273681709212104939588567455
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.