Dirichlet series
| L(s) = 1 | + 2·5-s − 4·17-s − 6·23-s + 6·25-s + 12·29-s − 28·31-s + 12·41-s − 18·43-s − 12·47-s + 16·49-s + 6·53-s − 10·59-s − 4·61-s + 6·67-s + 8·71-s − 8·73-s + 14·79-s − 8·85-s − 54·89-s + 60·97-s − 28·103-s + 16·107-s − 12·115-s + 48·121-s + 40·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.970·17-s − 1.25·23-s + 6/5·25-s + 2.22·29-s − 5.02·31-s + 1.87·41-s − 2.74·43-s − 1.75·47-s + 16/7·49-s + 0.824·53-s − 1.30·59-s − 0.512·61-s + 0.733·67-s + 0.949·71-s − 0.936·73-s + 1.57·79-s − 0.867·85-s − 5.72·89-s + 6.09·97-s − 2.75·103-s + 1.54·107-s − 1.11·115-s + 4.36·121-s + 3.57·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 3^{16} \cdot 19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.18973\times 10^{10}\) |
| Root analytic conductor: | \(4.67408\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 3^{16} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2731580982\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2731580982\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 19 | \( 1 - 4 T^{2} - 72 T^{3} + 258 T^{4} - 72 p T^{5} - 4 p^{2} T^{6} + p^{4} T^{8} \) | |
| good | 5 | \( 1 - 2 T - 2 T^{2} - 24 T^{3} + 56 T^{4} + 34 T^{5} + 444 T^{6} - 958 T^{7} - 641 T^{8} - 958 p T^{9} + 444 p^{2} T^{10} + 34 p^{3} T^{11} + 56 p^{4} T^{12} - 24 p^{5} T^{13} - 2 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 16 T^{2} + 110 T^{4} - 456 T^{6} + 1787 T^{8} - 456 p^{2} T^{10} + 110 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) | |
| 11 | \( 1 - 48 T^{2} + 1268 T^{4} - 2040 p T^{6} + 288234 T^{8} - 2040 p^{3} T^{10} + 1268 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} \) | |
| 13 | \( 1 + 2 p T^{2} + 353 T^{4} + 72 p T^{5} + 1338 T^{6} + 23256 T^{7} - 16012 T^{8} + 23256 p T^{9} + 1338 p^{2} T^{10} + 72 p^{4} T^{11} + 353 p^{4} T^{12} + 2 p^{7} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 + 4 T - 8 T^{2} + 216 T^{3} + 938 T^{4} - 1484 T^{5} + 1440 p T^{6} + 103676 T^{7} - 139085 T^{8} + 103676 p T^{9} + 1440 p^{3} T^{10} - 1484 p^{3} T^{11} + 938 p^{4} T^{12} + 216 p^{5} T^{13} - 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 6 T + 66 T^{2} + 324 T^{3} + 2252 T^{4} + 13578 T^{5} + 132 p^{2} T^{6} + 430122 T^{7} + 1727991 T^{8} + 430122 p T^{9} + 132 p^{4} T^{10} + 13578 p^{3} T^{11} + 2252 p^{4} T^{12} + 324 p^{5} T^{13} + 66 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 12 T + 108 T^{2} - 720 T^{3} + 3386 T^{4} - 5580 T^{5} - 54960 T^{6} + 730068 T^{7} - 4890669 T^{8} + 730068 p T^{9} - 54960 p^{2} T^{10} - 5580 p^{3} T^{11} + 3386 p^{4} T^{12} - 720 p^{5} T^{13} + 108 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( ( 1 + 14 T + 134 T^{2} + 1020 T^{3} + 6341 T^{4} + 1020 p T^{5} + 134 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 37 | \( 1 - 4 p T^{2} + 11522 T^{4} - 649824 T^{6} + 27865931 T^{8} - 649824 p^{2} T^{10} + 11522 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) | |
| 41 | \( 1 - 12 T + 152 T^{2} - 1248 T^{3} + 9322 T^{4} - 52164 T^{5} + 242912 T^{6} - 1054428 T^{7} + 4843411 T^{8} - 1054428 p T^{9} + 242912 p^{2} T^{10} - 52164 p^{3} T^{11} + 9322 p^{4} T^{12} - 1248 p^{5} T^{13} + 152 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 18 T + 214 T^{2} + 1908 T^{3} + 12631 T^{4} + 86184 T^{5} + 13978 p T^{6} + 4361166 T^{7} + 31873036 T^{8} + 4361166 p T^{9} + 13978 p^{3} T^{10} + 86184 p^{3} T^{11} + 12631 p^{4} T^{12} + 1908 p^{5} T^{13} + 214 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 12 T + 168 T^{2} + 1440 T^{3} + 12650 T^{4} + 93732 T^{5} + 604320 T^{6} + 4414788 T^{7} + 26589651 T^{8} + 4414788 p T^{9} + 604320 p^{2} T^{10} + 93732 p^{3} T^{11} + 12650 p^{4} T^{12} + 1440 p^{5} T^{13} + 168 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 6 T + 210 T^{2} - 1188 T^{3} + 25640 T^{4} - 131370 T^{5} + 2109660 T^{6} - 9700302 T^{7} + 129335247 T^{8} - 9700302 p T^{9} + 2109660 p^{2} T^{10} - 131370 p^{3} T^{11} + 25640 p^{4} T^{12} - 1188 p^{5} T^{13} + 210 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 10 T - 38 T^{2} - 936 T^{3} - 5620 T^{4} - 33554 T^{5} - 187092 T^{6} + 3151970 T^{7} + 54740455 T^{8} + 3151970 p T^{9} - 187092 p^{2} T^{10} - 33554 p^{3} T^{11} - 5620 p^{4} T^{12} - 936 p^{5} T^{13} - 38 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 4 T - 90 T^{2} - 40 T^{3} + 3557 T^{4} - 15264 T^{5} + 70462 T^{6} + 1051636 T^{7} - 6098868 T^{8} + 1051636 p T^{9} + 70462 p^{2} T^{10} - 15264 p^{3} T^{11} + 3557 p^{4} T^{12} - 40 p^{5} T^{13} - 90 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 6 T - 26 T^{2} - 732 T^{3} + 6427 T^{4} + 5184 T^{5} + 754582 T^{6} - 5240466 T^{7} - 15663692 T^{8} - 5240466 p T^{9} + 754582 p^{2} T^{10} + 5184 p^{3} T^{11} + 6427 p^{4} T^{12} - 732 p^{5} T^{13} - 26 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 8 T - 152 T^{2} + 1440 T^{3} + 12602 T^{4} - 112496 T^{5} - 831456 T^{6} + 3141224 T^{7} + 64254979 T^{8} + 3141224 p T^{9} - 831456 p^{2} T^{10} - 112496 p^{3} T^{11} + 12602 p^{4} T^{12} + 1440 p^{5} T^{13} - 152 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 8 T - 190 T^{2} - 704 T^{3} + 27773 T^{4} + 34552 T^{5} - 2773302 T^{6} - 1497216 T^{7} + 206564620 T^{8} - 1497216 p T^{9} - 2773302 p^{2} T^{10} + 34552 p^{3} T^{11} + 27773 p^{4} T^{12} - 704 p^{5} T^{13} - 190 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 14 T - 66 T^{2} + 644 T^{3} + 11855 T^{4} + 9744 T^{5} - 1436426 T^{6} + 2094862 T^{7} + 66521484 T^{8} + 2094862 p T^{9} - 1436426 p^{2} T^{10} + 9744 p^{3} T^{11} + 11855 p^{4} T^{12} + 644 p^{5} T^{13} - 66 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 160 T^{2} + 14188 T^{4} - 1709728 T^{6} + 189871366 T^{8} - 1709728 p^{2} T^{10} + 14188 p^{4} T^{12} - 160 p^{6} T^{14} + p^{8} T^{16} \) | |
| 89 | \( 1 + 54 T + 1530 T^{2} + 30132 T^{3} + 461336 T^{4} + 5878446 T^{5} + 65654868 T^{6} + 671327862 T^{7} + 6482561007 T^{8} + 671327862 p T^{9} + 65654868 p^{2} T^{10} + 5878446 p^{3} T^{11} + 461336 p^{4} T^{12} + 30132 p^{5} T^{13} + 1530 p^{6} T^{14} + 54 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 60 T + 2012 T^{2} - 48720 T^{3} + 939494 T^{4} - 15092220 T^{5} + 207598272 T^{6} - 2484380220 T^{7} + 26090169059 T^{8} - 2484380220 p T^{9} + 207598272 p^{2} T^{10} - 15092220 p^{3} T^{11} + 939494 p^{4} T^{12} - 48720 p^{5} T^{13} + 2012 p^{6} T^{14} - 60 p^{7} T^{15} + p^{8} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.66388070061227990517769864239, −3.56216781480176212391091190348, −3.34499124626304166997373552879, −3.30878352087923875223267373912, −3.24148131880346053412105978015, −3.23894067615354897212020300741, −2.88769207937753967984874605313, −2.76967888627357160950112082468, −2.70508397674769239452114215880, −2.53282168173581213352506733978, −2.42024737294830341340585234965, −2.29098051959717054347613144955, −2.08898882574972479503132030042, −2.08330482411358380866973197947, −1.87702059563315465753239426091, −1.79416316256704264342870120879, −1.72739571266745629028530828107, −1.58697156926495302727022364876, −1.25744770833530623751667601261, −1.17326941397937342011628409687, −1.06194013664209475663251511265, −0.73262425323182233536924110290, −0.65666153284866474273408258803, −0.28216678945245805570632416346, −0.05343371462235111268870016426, 0.05343371462235111268870016426, 0.28216678945245805570632416346, 0.65666153284866474273408258803, 0.73262425323182233536924110290, 1.06194013664209475663251511265, 1.17326941397937342011628409687, 1.25744770833530623751667601261, 1.58697156926495302727022364876, 1.72739571266745629028530828107, 1.79416316256704264342870120879, 1.87702059563315465753239426091, 2.08330482411358380866973197947, 2.08898882574972479503132030042, 2.29098051959717054347613144955, 2.42024737294830341340585234965, 2.53282168173581213352506733978, 2.70508397674769239452114215880, 2.76967888627357160950112082468, 2.88769207937753967984874605313, 3.23894067615354897212020300741, 3.24148131880346053412105978015, 3.30878352087923875223267373912, 3.34499124626304166997373552879, 3.56216781480176212391091190348, 3.66388070061227990517769864239
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.