Dirichlet series
| L(s) = 1 | − 3·3-s − 6·4-s − 4·5-s − 5·7-s − 8-s − 7·9-s − 7·11-s + 18·12-s + 2·13-s + 12·15-s + 14·16-s − 15·17-s − 24·19-s + 24·20-s + 15·21-s − 10·23-s + 3·24-s − 17·25-s + 34·27-s + 30·28-s + 29-s − 13·31-s + 6·32-s + 21·33-s + 20·35-s + 42·36-s − 16·37-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 3·4-s − 1.78·5-s − 1.88·7-s − 0.353·8-s − 7/3·9-s − 2.11·11-s + 5.19·12-s + 0.554·13-s + 3.09·15-s + 7/2·16-s − 3.63·17-s − 5.50·19-s + 5.36·20-s + 3.27·21-s − 2.08·23-s + 0.612·24-s − 3.39·25-s + 6.54·27-s + 5.66·28-s + 0.185·29-s − 2.33·31-s + 1.06·32-s + 3.65·33-s + 3.38·35-s + 7·36-s − 2.63·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(461^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(461^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(461^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(9158.96\) |
| Root analytic conductor: | \(1.91862\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 461^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
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 | 461 | \( ( 1 + T )^{7} \) |
| good | 2 | \( 1 + 3 p T^{2} + T^{3} + 11 p T^{4} + 3 p T^{5} + 7 p^{3} T^{6} + 17 T^{7} + 7 p^{4} T^{8} + 3 p^{3} T^{9} + 11 p^{4} T^{10} + p^{4} T^{11} + 3 p^{6} T^{12} + p^{7} T^{14} \) |
| 3 | \( 1 + p T + 16 T^{2} + 35 T^{3} + 106 T^{4} + 185 T^{5} + 425 T^{6} + 641 T^{7} + 425 p T^{8} + 185 p^{2} T^{9} + 106 p^{3} T^{10} + 35 p^{4} T^{11} + 16 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) | |
| 5 | \( 1 + 4 T + 33 T^{2} + 94 T^{3} + 442 T^{4} + 971 T^{5} + 3383 T^{6} + 6011 T^{7} + 3383 p T^{8} + 971 p^{2} T^{9} + 442 p^{3} T^{10} + 94 p^{4} T^{11} + 33 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 + 5 T + 32 T^{2} + 106 T^{3} + 451 T^{4} + 1208 T^{5} + 4185 T^{6} + 9665 T^{7} + 4185 p T^{8} + 1208 p^{2} T^{9} + 451 p^{3} T^{10} + 106 p^{4} T^{11} + 32 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 7 T + 67 T^{2} + 343 T^{3} + 1821 T^{4} + 7421 T^{5} + 28892 T^{6} + 98889 T^{7} + 28892 p T^{8} + 7421 p^{2} T^{9} + 1821 p^{3} T^{10} + 343 p^{4} T^{11} + 67 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 2 T + 45 T^{2} - 84 T^{3} + 1016 T^{4} - 1506 T^{5} + 16736 T^{6} - 19595 T^{7} + 16736 p T^{8} - 1506 p^{2} T^{9} + 1016 p^{3} T^{10} - 84 p^{4} T^{11} + 45 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 15 T + 177 T^{2} + 1451 T^{3} + 10215 T^{4} + 58893 T^{5} + 300922 T^{6} + 77205 p T^{7} + 300922 p T^{8} + 58893 p^{2} T^{9} + 10215 p^{3} T^{10} + 1451 p^{4} T^{11} + 177 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 24 T + 320 T^{2} + 3024 T^{3} + 22746 T^{4} + 143173 T^{5} + 774286 T^{6} + 3622399 T^{7} + 774286 p T^{8} + 143173 p^{2} T^{9} + 22746 p^{3} T^{10} + 3024 p^{4} T^{11} + 320 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 10 T + 129 T^{2} + 882 T^{3} + 6743 T^{4} + 36366 T^{5} + 213736 T^{6} + 980877 T^{7} + 213736 p T^{8} + 36366 p^{2} T^{9} + 6743 p^{3} T^{10} + 882 p^{4} T^{11} + 129 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - T + 103 T^{2} + 123 T^{3} + 4105 T^{4} + 19857 T^{5} + 93174 T^{6} + 898923 T^{7} + 93174 p T^{8} + 19857 p^{2} T^{9} + 4105 p^{3} T^{10} + 123 p^{4} T^{11} + 103 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 13 T + 6 p T^{2} + 1222 T^{3} + 8821 T^{4} + 26026 T^{5} + 129831 T^{6} + 45327 T^{7} + 129831 p T^{8} + 26026 p^{2} T^{9} + 8821 p^{3} T^{10} + 1222 p^{4} T^{11} + 6 p^{6} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 16 T + 206 T^{2} + 1886 T^{3} + 16454 T^{4} + 117499 T^{5} + 840742 T^{6} + 5169341 T^{7} + 840742 p T^{8} + 117499 p^{2} T^{9} + 16454 p^{3} T^{10} + 1886 p^{4} T^{11} + 206 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 4 T + 153 T^{2} - 482 T^{3} + 11848 T^{4} - 33900 T^{5} + 668098 T^{6} - 1712557 T^{7} + 668098 p T^{8} - 33900 p^{2} T^{9} + 11848 p^{3} T^{10} - 482 p^{4} T^{11} + 153 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 31 T + 624 T^{2} + 8873 T^{3} + 102662 T^{4} + 978591 T^{5} + 8038859 T^{6} + 56462697 T^{7} + 8038859 p T^{8} + 978591 p^{2} T^{9} + 102662 p^{3} T^{10} + 8873 p^{4} T^{11} + 624 p^{5} T^{12} + 31 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 16 T + 289 T^{2} - 3177 T^{3} + 34381 T^{4} - 295032 T^{5} + 51447 p T^{6} - 17004597 T^{7} + 51447 p^{2} T^{8} - 295032 p^{2} T^{9} + 34381 p^{3} T^{10} - 3177 p^{4} T^{11} + 289 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 29 T + 660 T^{2} - 10059 T^{3} + 131393 T^{4} - 1365774 T^{5} + 12588841 T^{6} - 96840615 T^{7} + 12588841 p T^{8} - 1365774 p^{2} T^{9} + 131393 p^{3} T^{10} - 10059 p^{4} T^{11} + 660 p^{5} T^{12} - 29 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 14 T + 304 T^{2} - 3252 T^{3} + 42573 T^{4} - 361684 T^{5} + 3647159 T^{6} - 25715899 T^{7} + 3647159 p T^{8} - 361684 p^{2} T^{9} + 42573 p^{3} T^{10} - 3252 p^{4} T^{11} + 304 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 15 T + 112 T^{2} + 280 T^{3} - 810 T^{4} + 13794 T^{5} + 510577 T^{6} + 5604645 T^{7} + 510577 p T^{8} + 13794 p^{2} T^{9} - 810 p^{3} T^{10} + 280 p^{4} T^{11} + 112 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 34 T + 794 T^{2} + 13745 T^{3} + 195609 T^{4} + 2315074 T^{5} + 23615394 T^{6} + 207274103 T^{7} + 23615394 p T^{8} + 2315074 p^{2} T^{9} + 195609 p^{3} T^{10} + 13745 p^{4} T^{11} + 794 p^{5} T^{12} + 34 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 8 T + 283 T^{2} + 2008 T^{3} + 44120 T^{4} + 286468 T^{5} + 4481806 T^{6} + 24695519 T^{7} + 4481806 p T^{8} + 286468 p^{2} T^{9} + 44120 p^{3} T^{10} + 2008 p^{4} T^{11} + 283 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 11 T + 253 T^{2} - 2848 T^{3} + 37670 T^{4} - 365771 T^{5} + 4028917 T^{6} - 31720673 T^{7} + 4028917 p T^{8} - 365771 p^{2} T^{9} + 37670 p^{3} T^{10} - 2848 p^{4} T^{11} + 253 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 11 T + 489 T^{2} + 4106 T^{3} + 102619 T^{4} + 681097 T^{5} + 12519577 T^{6} + 67205839 T^{7} + 12519577 p T^{8} + 681097 p^{2} T^{9} + 102619 p^{3} T^{10} + 4106 p^{4} T^{11} + 489 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 19 T + 457 T^{2} - 6508 T^{3} + 97747 T^{4} - 1124813 T^{5} + 151073 p T^{6} - 116750523 T^{7} + 151073 p^{2} T^{8} - 1124813 p^{2} T^{9} + 97747 p^{3} T^{10} - 6508 p^{4} T^{11} + 457 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 17 T + 400 T^{2} + 5869 T^{3} + 78106 T^{4} + 931002 T^{5} + 10007624 T^{6} + 96731675 T^{7} + 10007624 p T^{8} + 931002 p^{2} T^{9} + 78106 p^{3} T^{10} + 5869 p^{4} T^{11} + 400 p^{5} T^{12} + 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 16 T + 483 T^{2} + 6135 T^{3} + 111308 T^{4} + 1194375 T^{5} + 16154400 T^{6} + 144186993 T^{7} + 16154400 p T^{8} + 1194375 p^{2} T^{9} + 111308 p^{3} T^{10} + 6135 p^{4} T^{11} + 483 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.82147425545866907979065940001, −5.81287056851735308655971926810, −5.67190869245820729404297567468, −5.39669236442342635306653476415, −5.33915728244087688651428806725, −4.98352485983565559016584454825, −4.87635784350864918757449253108, −4.87002047477699864403793860198, −4.84213960827088631623601165329, −4.31245037122871056499774780543, −4.30069465437673760628385649622, −4.11611340946615019927805964424, −4.00282435976119805265566597252, −3.91701219907871463752019511262, −3.72333984712001881819193947748, −3.71652934888607826804129039755, −3.56035795865822183869774783228, −3.13182171743925541644668761694, −2.91543546236412864928068728154, −2.72362618454792296829969135173, −2.37146940075596806747051709444, −2.22658295472661922849686515910, −2.16607317643738757358194060748, −2.00990915162398642415578151739, −1.69648623595227407696790317799, 0, 0, 0, 0, 0, 0, 0, 1.69648623595227407696790317799, 2.00990915162398642415578151739, 2.16607317643738757358194060748, 2.22658295472661922849686515910, 2.37146940075596806747051709444, 2.72362618454792296829969135173, 2.91543546236412864928068728154, 3.13182171743925541644668761694, 3.56035795865822183869774783228, 3.71652934888607826804129039755, 3.72333984712001881819193947748, 3.91701219907871463752019511262, 4.00282435976119805265566597252, 4.11611340946615019927805964424, 4.30069465437673760628385649622, 4.31245037122871056499774780543, 4.84213960827088631623601165329, 4.87002047477699864403793860198, 4.87635784350864918757449253108, 4.98352485983565559016584454825, 5.33915728244087688651428806725, 5.39669236442342635306653476415, 5.67190869245820729404297567468, 5.81287056851735308655971926810, 5.82147425545866907979065940001
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.