Dirichlet series
| L(s) = 1 | − 5·3-s − 8·5-s + 8·7-s + 6·9-s + 6·11-s − 13·13-s + 40·15-s − 2·17-s − 14·19-s − 40·21-s + 6·23-s + 36·25-s + 13·27-s − 7·29-s − 18·31-s − 30·33-s − 64·35-s + 2·37-s + 65·39-s − 18·41-s − 8·43-s − 48·45-s − 47-s + 36·49-s + 10·51-s + 5·53-s − 48·55-s + ⋯ |
| L(s) = 1 | − 2.88·3-s − 3.57·5-s + 3.02·7-s + 2·9-s + 1.80·11-s − 3.60·13-s + 10.3·15-s − 0.485·17-s − 3.21·19-s − 8.72·21-s + 1.25·23-s + 36/5·25-s + 2.50·27-s − 1.29·29-s − 3.23·31-s − 5.22·33-s − 10.8·35-s + 0.328·37-s + 10.4·39-s − 2.81·41-s − 1.21·43-s − 7.15·45-s − 0.145·47-s + 36/7·49-s + 1.40·51-s + 0.686·53-s − 6.47·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{8} \cdot 43^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{8} \cdot 43^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 5^{8} \cdot 7^{8} \cdot 43^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.85094\times 10^{13}\) |
| Root analytic conductor: | \(6.93324\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{16} \cdot 5^{8} \cdot 7^{8} \cdot 43^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | \( ( 1 + T )^{8} \) | |
| 7 | \( ( 1 - T )^{8} \) | |
| 43 | \( ( 1 + T )^{8} \) | |
| good | 3 | \( 1 + 5 T + 19 T^{2} + 52 T^{3} + 44 p T^{4} + 289 T^{5} + 614 T^{6} + 391 p T^{7} + 718 p T^{8} + 391 p^{2} T^{9} + 614 p^{2} T^{10} + 289 p^{3} T^{11} + 44 p^{5} T^{12} + 52 p^{5} T^{13} + 19 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 - 6 T + 76 T^{2} - 358 T^{3} + 233 p T^{4} - 9846 T^{5} + 51408 T^{6} - 163971 T^{7} + 683819 T^{8} - 163971 p T^{9} + 51408 p^{2} T^{10} - 9846 p^{3} T^{11} + 233 p^{5} T^{12} - 358 p^{5} T^{13} + 76 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + p T + 134 T^{2} + 1017 T^{3} + 6567 T^{4} + 35777 T^{5} + 173178 T^{6} + 735641 T^{7} + 216511 p T^{8} + 735641 p T^{9} + 173178 p^{2} T^{10} + 35777 p^{3} T^{11} + 6567 p^{4} T^{12} + 1017 p^{5} T^{13} + 134 p^{6} T^{14} + p^{8} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 2 T + 92 T^{2} + 104 T^{3} + 3902 T^{4} + 2316 T^{5} + 106465 T^{6} + 2171 p T^{7} + 2101043 T^{8} + 2171 p^{2} T^{9} + 106465 p^{2} T^{10} + 2316 p^{3} T^{11} + 3902 p^{4} T^{12} + 104 p^{5} T^{13} + 92 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 14 T + 200 T^{2} + 1785 T^{3} + 15075 T^{4} + 98910 T^{5} + 607054 T^{6} + 3087698 T^{7} + 14666288 T^{8} + 3087698 p T^{9} + 607054 p^{2} T^{10} + 98910 p^{3} T^{11} + 15075 p^{4} T^{12} + 1785 p^{5} T^{13} + 200 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 6 T + 118 T^{2} - 527 T^{3} + 6587 T^{4} - 24353 T^{5} + 245438 T^{6} - 788568 T^{7} + 6633435 T^{8} - 788568 p T^{9} + 245438 p^{2} T^{10} - 24353 p^{3} T^{11} + 6587 p^{4} T^{12} - 527 p^{5} T^{13} + 118 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 7 T + 138 T^{2} + 924 T^{3} + 9882 T^{4} + 62935 T^{5} + 470976 T^{6} + 2698771 T^{7} + 15971198 T^{8} + 2698771 p T^{9} + 470976 p^{2} T^{10} + 62935 p^{3} T^{11} + 9882 p^{4} T^{12} + 924 p^{5} T^{13} + 138 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 18 T + 271 T^{2} + 2957 T^{3} + 28463 T^{4} + 231611 T^{5} + 1702274 T^{6} + 11014636 T^{7} + 65084849 T^{8} + 11014636 p T^{9} + 1702274 p^{2} T^{10} + 231611 p^{3} T^{11} + 28463 p^{4} T^{12} + 2957 p^{5} T^{13} + 271 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 2 T + 137 T^{2} - 411 T^{3} + 8539 T^{4} - 38013 T^{5} + 337663 T^{6} - 2106519 T^{7} + 11780298 T^{8} - 2106519 p T^{9} + 337663 p^{2} T^{10} - 38013 p^{3} T^{11} + 8539 p^{4} T^{12} - 411 p^{5} T^{13} + 137 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 18 T + 324 T^{2} + 3411 T^{3} + 36718 T^{4} + 287376 T^{5} + 2385997 T^{6} + 15504512 T^{7} + 110758693 T^{8} + 15504512 p T^{9} + 2385997 p^{2} T^{10} + 287376 p^{3} T^{11} + 36718 p^{4} T^{12} + 3411 p^{5} T^{13} + 324 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + T + 124 T^{2} - 144 T^{3} + 9834 T^{4} - 18049 T^{5} + 570894 T^{6} - 1480695 T^{7} + 27868278 T^{8} - 1480695 p T^{9} + 570894 p^{2} T^{10} - 18049 p^{3} T^{11} + 9834 p^{4} T^{12} - 144 p^{5} T^{13} + 124 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 5 T + 253 T^{2} - 729 T^{3} + 31220 T^{4} - 60127 T^{5} + 2641475 T^{6} - 4010375 T^{7} + 163372773 T^{8} - 4010375 p T^{9} + 2641475 p^{2} T^{10} - 60127 p^{3} T^{11} + 31220 p^{4} T^{12} - 729 p^{5} T^{13} + 253 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 12 T + 404 T^{2} + 4430 T^{3} + 75243 T^{4} + 727629 T^{5} + 8367731 T^{6} + 68942594 T^{7} + 605274118 T^{8} + 68942594 p T^{9} + 8367731 p^{2} T^{10} + 727629 p^{3} T^{11} + 75243 p^{4} T^{12} + 4430 p^{5} T^{13} + 404 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 23 T + 527 T^{2} + 7448 T^{3} + 102451 T^{4} + 1080619 T^{5} + 11232330 T^{6} + 96039059 T^{7} + 819014312 T^{8} + 96039059 p T^{9} + 11232330 p^{2} T^{10} + 1080619 p^{3} T^{11} + 102451 p^{4} T^{12} + 7448 p^{5} T^{13} + 527 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 8 T + 356 T^{2} - 2041 T^{3} + 57788 T^{4} - 256400 T^{5} + 90923 p T^{6} - 22578502 T^{7} + 471313787 T^{8} - 22578502 p T^{9} + 90923 p^{3} T^{10} - 256400 p^{3} T^{11} + 57788 p^{4} T^{12} - 2041 p^{5} T^{13} + 356 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 20 T + 458 T^{2} - 7071 T^{3} + 102978 T^{4} - 1218643 T^{5} + 13711016 T^{6} - 131516341 T^{7} + 1185868706 T^{8} - 131516341 p T^{9} + 13711016 p^{2} T^{10} - 1218643 p^{3} T^{11} + 102978 p^{4} T^{12} - 7071 p^{5} T^{13} + 458 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 4 T + 318 T^{2} + 1599 T^{3} + 53216 T^{4} + 279135 T^{5} + 6127590 T^{6} + 29626981 T^{7} + 519439350 T^{8} + 29626981 p T^{9} + 6127590 p^{2} T^{10} + 279135 p^{3} T^{11} + 53216 p^{4} T^{12} + 1599 p^{5} T^{13} + 318 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 24 T + 428 T^{2} - 5401 T^{3} + 58561 T^{4} - 523882 T^{5} + 4386694 T^{6} - 33738552 T^{7} + 294541744 T^{8} - 33738552 p T^{9} + 4386694 p^{2} T^{10} - 523882 p^{3} T^{11} + 58561 p^{4} T^{12} - 5401 p^{5} T^{13} + 428 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 14 T + 422 T^{2} + 4628 T^{3} + 84131 T^{4} + 743218 T^{5} + 10567132 T^{6} + 79704611 T^{7} + 986225191 T^{8} + 79704611 p T^{9} + 10567132 p^{2} T^{10} + 743218 p^{3} T^{11} + 84131 p^{4} T^{12} + 4628 p^{5} T^{13} + 422 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 21 T + 485 T^{2} + 8163 T^{3} + 123068 T^{4} + 1605502 T^{5} + 19334031 T^{6} + 205274839 T^{7} + 2063077876 T^{8} + 205274839 p T^{9} + 19334031 p^{2} T^{10} + 1605502 p^{3} T^{11} + 123068 p^{4} T^{12} + 8163 p^{5} T^{13} + 485 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 7 T + 509 T^{2} - 3820 T^{3} + 130664 T^{4} - 959407 T^{5} + 21541709 T^{6} - 144045116 T^{7} + 2477897747 T^{8} - 144045116 p T^{9} + 21541709 p^{2} T^{10} - 959407 p^{3} T^{11} + 130664 p^{4} T^{12} - 3820 p^{5} T^{13} + 509 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.74770155124924106237552433954, −3.50248815811167954107190795503, −3.46406589491568328961640192870, −3.43142063992428002218479159170, −3.42374690182397866385737783164, −3.24597018355771638923011014733, −3.13549172609169615072120733897, −3.10222773977841191498270757047, −3.06697614649736314138713783372, −2.47724057821813098802795455920, −2.46945784740593001008180104884, −2.46500979569571257468403309597, −2.29372168732297806358589965032, −2.27390950640659280246223182992, −2.11621690308810437949959828443, −2.11366231838460113666240270228, −2.07947209179749840566566308697, −1.75491752403375681419360743523, −1.57039528009365363449734863856, −1.38164906636104684433680638096, −1.21073843351076410990259723387, −1.11443423402225187735254475061, −1.02617105224621104663277151441, −1.01618230572415559550786860081, −0.998626947109694490440724505505, 0, 0, 0, 0, 0, 0, 0, 0, 0.998626947109694490440724505505, 1.01618230572415559550786860081, 1.02617105224621104663277151441, 1.11443423402225187735254475061, 1.21073843351076410990259723387, 1.38164906636104684433680638096, 1.57039528009365363449734863856, 1.75491752403375681419360743523, 2.07947209179749840566566308697, 2.11366231838460113666240270228, 2.11621690308810437949959828443, 2.27390950640659280246223182992, 2.29372168732297806358589965032, 2.46500979569571257468403309597, 2.46945784740593001008180104884, 2.47724057821813098802795455920, 3.06697614649736314138713783372, 3.10222773977841191498270757047, 3.13549172609169615072120733897, 3.24597018355771638923011014733, 3.42374690182397866385737783164, 3.43142063992428002218479159170, 3.46406589491568328961640192870, 3.50248815811167954107190795503, 3.74770155124924106237552433954
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.