Dirichlet series
| L(s) = 1 | − 3-s − 7·7-s − 7·9-s − 7·11-s + 10·13-s − 3·17-s − 7·19-s + 7·21-s − 14·23-s + 4·27-s − 11·29-s + 11·31-s + 7·33-s + 13·37-s − 10·39-s + 4·41-s − 2·43-s − 22·47-s + 8·49-s + 3·51-s − 3·53-s + 7·57-s − 26·59-s − 5·61-s + 49·63-s − 14·67-s + 14·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2.64·7-s − 7/3·9-s − 2.11·11-s + 2.77·13-s − 0.727·17-s − 1.60·19-s + 1.52·21-s − 2.91·23-s + 0.769·27-s − 2.04·29-s + 1.97·31-s + 1.21·33-s + 2.13·37-s − 1.60·39-s + 0.624·41-s − 0.304·43-s − 3.20·47-s + 8/7·49-s + 0.420·51-s − 0.412·53-s + 0.927·57-s − 3.38·59-s − 0.640·61-s + 6.17·63-s − 1.71·67-s + 1.68·69-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{35} \cdot 5^{14} \cdot 11^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{35} \cdot 5^{14} \cdot 11^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{35} \cdot 5^{14} \cdot 11^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(8.45898\times 10^{12}\) |
| Root analytic conductor: | \(8.38262\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 2^{35} \cdot 5^{14} \cdot 11^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
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 | 2 | \( 1 \) |
| 5 | \( 1 \) | |
| 11 | \( ( 1 + T )^{7} \) | |
| good | 3 | \( 1 + T + 8 T^{2} + 11 T^{3} + 40 T^{4} + 7 p^{2} T^{5} + 149 T^{6} + 218 T^{7} + 149 p T^{8} + 7 p^{4} T^{9} + 40 p^{3} T^{10} + 11 p^{4} T^{11} + 8 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 + p T + 41 T^{2} + 158 T^{3} + 517 T^{4} + 1353 T^{5} + 3469 T^{6} + 8388 T^{7} + 3469 p T^{8} + 1353 p^{2} T^{9} + 517 p^{3} T^{10} + 158 p^{4} T^{11} + 41 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 10 T + 7 p T^{2} - 540 T^{3} + 3101 T^{4} - 13990 T^{5} + 62047 T^{6} - 226184 T^{7} + 62047 p T^{8} - 13990 p^{2} T^{9} + 3101 p^{3} T^{10} - 540 p^{4} T^{11} + 7 p^{6} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 3 T + 5 p T^{2} + 134 T^{3} + 3011 T^{4} + 1549 T^{5} + 65511 T^{6} + 4820 T^{7} + 65511 p T^{8} + 1549 p^{2} T^{9} + 3011 p^{3} T^{10} + 134 p^{4} T^{11} + 5 p^{6} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 7 T + 81 T^{2} + 458 T^{3} + 9 p^{2} T^{4} + 15649 T^{5} + 90585 T^{6} + 361036 T^{7} + 90585 p T^{8} + 15649 p^{2} T^{9} + 9 p^{5} T^{10} + 458 p^{4} T^{11} + 81 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 14 T + 182 T^{2} + 1534 T^{3} + 11968 T^{4} + 75234 T^{5} + 439571 T^{6} + 2189284 T^{7} + 439571 p T^{8} + 75234 p^{2} T^{9} + 11968 p^{3} T^{10} + 1534 p^{4} T^{11} + 182 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 11 T + 191 T^{2} + 1546 T^{3} + 15345 T^{4} + 97565 T^{5} + 705015 T^{6} + 3595212 T^{7} + 705015 p T^{8} + 97565 p^{2} T^{9} + 15345 p^{3} T^{10} + 1546 p^{4} T^{11} + 191 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 11 T + 176 T^{2} - 1659 T^{3} + 15282 T^{4} - 3599 p T^{5} + 775891 T^{6} - 4391506 T^{7} + 775891 p T^{8} - 3599 p^{3} T^{9} + 15282 p^{3} T^{10} - 1659 p^{4} T^{11} + 176 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 13 T + 236 T^{2} - 2015 T^{3} + 21274 T^{4} - 137199 T^{5} + 1107157 T^{6} - 5952378 T^{7} + 1107157 p T^{8} - 137199 p^{2} T^{9} + 21274 p^{3} T^{10} - 2015 p^{4} T^{11} + 236 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 4 T + 107 T^{2} - 712 T^{3} + 9905 T^{4} - 46716 T^{5} + 549331 T^{6} - 2640880 T^{7} + 549331 p T^{8} - 46716 p^{2} T^{9} + 9905 p^{3} T^{10} - 712 p^{4} T^{11} + 107 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 2 T + 185 T^{2} + 740 T^{3} + 17345 T^{4} + 75182 T^{5} + 1118161 T^{6} + 4026296 T^{7} + 1118161 p T^{8} + 75182 p^{2} T^{9} + 17345 p^{3} T^{10} + 740 p^{4} T^{11} + 185 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 22 T + 313 T^{2} + 2972 T^{3} + 25413 T^{4} + 198890 T^{5} + 1602285 T^{6} + 11395912 T^{7} + 1602285 p T^{8} + 198890 p^{2} T^{9} + 25413 p^{3} T^{10} + 2972 p^{4} T^{11} + 313 p^{5} T^{12} + 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 3 T + 3 p T^{2} + 170 T^{3} + 7865 T^{4} - 13947 T^{5} + 84095 T^{6} - 1548372 T^{7} + 84095 p T^{8} - 13947 p^{2} T^{9} + 7865 p^{3} T^{10} + 170 p^{4} T^{11} + 3 p^{6} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 26 T + 498 T^{2} + 5532 T^{3} + 47392 T^{4} + 210358 T^{5} + 436683 T^{6} - 3720440 T^{7} + 436683 p T^{8} + 210358 p^{2} T^{9} + 47392 p^{3} T^{10} + 5532 p^{4} T^{11} + 498 p^{5} T^{12} + 26 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 5 T + 295 T^{2} + 1174 T^{3} + 40857 T^{4} + 141587 T^{5} + 3618055 T^{6} + 10826740 T^{7} + 3618055 p T^{8} + 141587 p^{2} T^{9} + 40857 p^{3} T^{10} + 1174 p^{4} T^{11} + 295 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 14 T + 442 T^{2} + 4854 T^{3} + 83932 T^{4} + 739970 T^{5} + 9057687 T^{6} + 63992820 T^{7} + 9057687 p T^{8} + 739970 p^{2} T^{9} + 83932 p^{3} T^{10} + 4854 p^{4} T^{11} + 442 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 3 T + 140 T^{2} + 743 T^{3} + 6886 T^{4} + 33369 T^{5} + 457411 T^{6} - 605318 T^{7} + 457411 p T^{8} + 33369 p^{2} T^{9} + 6886 p^{3} T^{10} + 743 p^{4} T^{11} + 140 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 16 T + 307 T^{2} - 3392 T^{3} + 49769 T^{4} - 481168 T^{5} + 5226451 T^{6} - 40625728 T^{7} + 5226451 p T^{8} - 481168 p^{2} T^{9} + 49769 p^{3} T^{10} - 3392 p^{4} T^{11} + 307 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 32 T + 829 T^{2} - 14672 T^{3} + 224401 T^{4} - 2796192 T^{5} + 30891317 T^{6} - 290700256 T^{7} + 30891317 p T^{8} - 2796192 p^{2} T^{9} + 224401 p^{3} T^{10} - 14672 p^{4} T^{11} + 829 p^{5} T^{12} - 32 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 16 T + 481 T^{2} + 5448 T^{3} + 96433 T^{4} + 841328 T^{5} + 11443273 T^{6} + 82880240 T^{7} + 11443273 p T^{8} + 841328 p^{2} T^{9} + 96433 p^{3} T^{10} + 5448 p^{4} T^{11} + 481 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 11 T + 370 T^{2} - 4311 T^{3} + 76648 T^{4} - 766585 T^{5} + 115205 p T^{6} - 84526282 T^{7} + 115205 p^{2} T^{8} - 766585 p^{2} T^{9} + 76648 p^{3} T^{10} - 4311 p^{4} T^{11} + 370 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 8 T + 428 T^{2} + 3352 T^{3} + 96398 T^{4} + 662552 T^{5} + 13739509 T^{6} + 80902672 T^{7} + 13739509 p T^{8} + 662552 p^{2} T^{9} + 96398 p^{3} T^{10} + 3352 p^{4} T^{11} + 428 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.86128447970434357494194103395, −3.73448781941511461432814238442, −3.67814723843088838058856335112, −3.66398069144691450578939054739, −3.29775033791886761308655842502, −3.18365924644100948619854162641, −3.07928732900844689673664965725, −3.01827981945581899440859037264, −2.99970568905301492777063216668, −2.97589044533850115191564135264, −2.88191951595410049650872859896, −2.52035696562545371069765240693, −2.50380868612373960648107349104, −2.35126549756808256245809187318, −2.22113987699605925994888243700, −2.16044380149708711752732419716, −2.02092829188850561257367242453, −1.97159368496751232886459863713, −1.66720446432733552125707767326, −1.42365986284479187948885252449, −1.36865721181853560456578012092, −1.34608405413726450998899619835, −0.982972610017994503250450787375, −0.914084009810186263602321170110, −0.805781056845453089510322773812, 0, 0, 0, 0, 0, 0, 0, 0.805781056845453089510322773812, 0.914084009810186263602321170110, 0.982972610017994503250450787375, 1.34608405413726450998899619835, 1.36865721181853560456578012092, 1.42365986284479187948885252449, 1.66720446432733552125707767326, 1.97159368496751232886459863713, 2.02092829188850561257367242453, 2.16044380149708711752732419716, 2.22113987699605925994888243700, 2.35126549756808256245809187318, 2.50380868612373960648107349104, 2.52035696562545371069765240693, 2.88191951595410049650872859896, 2.97589044533850115191564135264, 2.99970568905301492777063216668, 3.01827981945581899440859037264, 3.07928732900844689673664965725, 3.18365924644100948619854162641, 3.29775033791886761308655842502, 3.66398069144691450578939054739, 3.67814723843088838058856335112, 3.73448781941511461432814238442, 3.86128447970434357494194103395
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.