Dirichlet series
| L(s) = 1 | + 8·9-s + 16·11-s + 8·13-s + 16·17-s + 16·19-s − 8·23-s − 8·27-s + 16·29-s + 16·37-s + 24·43-s − 16·47-s + 16·53-s − 24·61-s − 16·71-s − 8·73-s − 32·79-s + 40·81-s − 16·83-s − 8·89-s + 128·99-s − 32·101-s − 24·103-s − 16·107-s + 24·109-s + 48·113-s + 64·117-s + 128·121-s + ⋯ |
| L(s) = 1 | + 8/3·9-s + 4.82·11-s + 2.21·13-s + 3.88·17-s + 3.67·19-s − 1.66·23-s − 1.53·27-s + 2.97·29-s + 2.63·37-s + 3.65·43-s − 2.33·47-s + 2.19·53-s − 3.07·61-s − 1.89·71-s − 0.936·73-s − 3.60·79-s + 40/9·81-s − 1.75·83-s − 0.847·89-s + 12.8·99-s − 3.18·101-s − 2.36·103-s − 1.54·107-s + 2.29·109-s + 4.51·113-s + 5.91·117-s + 11.6·121-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{48} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(26818.9\) |
| Root analytic conductor: | \(1.89137\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{48} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(22.46710836\) |
| \(L(\frac12)\) | \(\approx\) | \(22.46710836\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + p^{4} T^{8} \) |
| 7 | \( 1 + T^{8} \) | |
| good | 3 | \( 1 - 8 T^{2} + 8 T^{3} + 8 p T^{4} - 56 T^{5} + 32 p T^{7} - 160 T^{8} + 32 p^{2} T^{9} - 56 p^{3} T^{11} + 8 p^{5} T^{12} + 8 p^{5} T^{13} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( 1 + p^{8} T^{16} \) | |
| 11 | \( 1 - 16 T + 128 T^{2} - 720 T^{3} + 3138 T^{4} - 10848 T^{5} + 31264 T^{6} - 82432 T^{7} + 240258 T^{8} - 82432 p T^{9} + 31264 p^{2} T^{10} - 10848 p^{3} T^{11} + 3138 p^{4} T^{12} - 720 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 8 T + 8 T^{2} + 136 T^{3} - 368 T^{4} - 136 p T^{5} + 8968 T^{6} + 8 p^{3} T^{7} - 190400 T^{8} + 8 p^{4} T^{9} + 8968 p^{2} T^{10} - 136 p^{4} T^{11} - 368 p^{4} T^{12} + 136 p^{5} T^{13} + 8 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 16 T + 128 T^{2} - 752 T^{3} + 4372 T^{4} - 25360 T^{5} + 128896 T^{6} - 560880 T^{7} + 2305638 T^{8} - 560880 p T^{9} + 128896 p^{2} T^{10} - 25360 p^{3} T^{11} + 4372 p^{4} T^{12} - 752 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 16 T + 136 T^{2} - 904 T^{3} + 5336 T^{4} - 28888 T^{5} + 140880 T^{6} - 633312 T^{7} + 2775648 T^{8} - 633312 p T^{9} + 140880 p^{2} T^{10} - 28888 p^{3} T^{11} + 5336 p^{4} T^{12} - 904 p^{5} T^{13} + 136 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 8 T + 28 T^{2} - 8 T^{3} - 440 T^{4} + 1720 T^{5} + 22996 T^{6} + 82376 T^{7} + 379598 T^{8} + 82376 p T^{9} + 22996 p^{2} T^{10} + 1720 p^{3} T^{11} - 440 p^{4} T^{12} - 8 p^{5} T^{13} + 28 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 16 T + 144 T^{2} - 1072 T^{3} + 7074 T^{4} - 43088 T^{5} + 286736 T^{6} - 1826672 T^{7} + 10349122 T^{8} - 1826672 p T^{9} + 286736 p^{2} T^{10} - 43088 p^{3} T^{11} + 7074 p^{4} T^{12} - 1072 p^{5} T^{13} + 144 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 8 T^{2} + 1804 T^{4} - 28680 T^{6} + 1332518 T^{8} - 28680 p^{2} T^{10} + 1804 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} \) | |
| 37 | \( 1 - 16 T + 1296 T^{3} - 4318 T^{4} - 56304 T^{5} + 373024 T^{6} + 1092016 T^{7} - 19195326 T^{8} + 1092016 p T^{9} + 373024 p^{2} T^{10} - 56304 p^{3} T^{11} - 4318 p^{4} T^{12} + 1296 p^{5} T^{13} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 4663486 T^{8} + p^{8} T^{16} \) | |
| 43 | \( 1 - 24 T + 268 T^{2} - 1800 T^{3} + 8166 T^{4} - 40200 T^{5} + 466732 T^{6} - 5606808 T^{7} + 45366722 T^{8} - 5606808 p T^{9} + 466732 p^{2} T^{10} - 40200 p^{3} T^{11} + 8166 p^{4} T^{12} - 1800 p^{5} T^{13} + 268 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + 5652 T^{4} + 49072 T^{5} + 344448 T^{6} + 28080 p T^{7} + 1702 p^{2} T^{8} + 28080 p^{2} T^{9} + 344448 p^{2} T^{10} + 49072 p^{3} T^{11} + 5652 p^{4} T^{12} + 16 p^{6} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 16 T + 132 T^{2} - 1472 T^{3} + 9158 T^{4} + 1664 T^{5} - 123132 T^{6} + 2884752 T^{7} - 41405566 T^{8} + 2884752 p T^{9} - 123132 p^{2} T^{10} + 1664 p^{3} T^{11} + 9158 p^{4} T^{12} - 1472 p^{5} T^{13} + 132 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 88 T^{2} - 792 T^{3} + 6744 T^{4} - 44344 T^{5} + 587344 T^{6} - 4859600 T^{7} + 17115104 T^{8} - 4859600 p T^{9} + 587344 p^{2} T^{10} - 44344 p^{3} T^{11} + 6744 p^{4} T^{12} - 792 p^{5} T^{13} + 88 p^{6} T^{14} + p^{8} T^{16} \) | |
| 61 | \( 1 + 24 T + 344 T^{2} + 4152 T^{3} + 39472 T^{4} + 312200 T^{5} + 2191336 T^{6} + 14408344 T^{7} + 104168512 T^{8} + 14408344 p T^{9} + 2191336 p^{2} T^{10} + 312200 p^{3} T^{11} + 39472 p^{4} T^{12} + 4152 p^{5} T^{13} + 344 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 36 T^{2} - 816 T^{3} + 5510 T^{4} + 3264 T^{5} + 747492 T^{6} - 4213008 T^{7} + 8734466 T^{8} - 4213008 p T^{9} + 747492 p^{2} T^{10} + 3264 p^{3} T^{11} + 5510 p^{4} T^{12} - 816 p^{5} T^{13} + 36 p^{6} T^{14} + p^{8} T^{16} \) | |
| 71 | \( 1 + 16 T + 96 T^{2} + 688 T^{3} + 7424 T^{4} + 111184 T^{5} + 995680 T^{6} + 3963120 T^{7} + 19042178 T^{8} + 3963120 p T^{9} + 995680 p^{2} T^{10} + 111184 p^{3} T^{11} + 7424 p^{4} T^{12} + 688 p^{5} T^{13} + 96 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 8 T + 24 T^{2} + 24 T^{3} - 32 T^{4} + 39640 T^{5} + 162360 T^{6} - 2485368 T^{7} - 34493598 T^{8} - 2485368 p T^{9} + 162360 p^{2} T^{10} + 39640 p^{3} T^{11} - 32 p^{4} T^{12} + 24 p^{5} T^{13} + 24 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 32 T + 512 T^{2} + 6048 T^{3} + 63008 T^{4} + 650336 T^{5} + 6839808 T^{6} + 70922720 T^{7} + 675882498 T^{8} + 70922720 p T^{9} + 6839808 p^{2} T^{10} + 650336 p^{3} T^{11} + 63008 p^{4} T^{12} + 6048 p^{5} T^{13} + 512 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 16 T + 248 T^{2} + 3656 T^{3} + 44120 T^{4} + 501288 T^{5} + 5260928 T^{6} + 53087920 T^{7} + 503492960 T^{8} + 53087920 p T^{9} + 5260928 p^{2} T^{10} + 501288 p^{3} T^{11} + 44120 p^{4} T^{12} + 3656 p^{5} T^{13} + 248 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 8 T + 168 T^{2} - 504 T^{3} + 352 T^{4} - 205560 T^{5} + 219272 T^{6} - 108152 p T^{7} + 113913698 T^{8} - 108152 p^{2} T^{9} + 219272 p^{2} T^{10} - 205560 p^{3} T^{11} + 352 p^{4} T^{12} - 504 p^{5} T^{13} + 168 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 712 T^{2} + 227468 T^{4} - 42558264 T^{6} + 5098416678 T^{8} - 42558264 p^{2} T^{10} + 227468 p^{4} T^{12} - 712 p^{6} T^{14} + p^{8} T^{16} \) | |
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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
−4.74310753747249420253072960649, −4.61706280632551662549072666168, −4.60992542254289008057490805557, −4.37970940487822485603434622975, −4.31638421287605068385708901979, −4.17988633345188948066322439203, −4.03793193164359412052108765761, −3.94222125156778967567481107005, −3.59538630973157184628621057825, −3.55107276389428130898270893044, −3.49659974432721453271525068978, −3.39222756409331222170147485735, −3.30902955590238593155924983351, −3.07571440787870853012824560642, −2.85014966305997503179814267282, −2.43469467811386696489074722890, −2.42679756532740297964386241743, −2.28163051032638861212547266899, −1.53848504184469208789448386843, −1.40221953573734773365907850071, −1.33468046563298565496915827435, −1.21023025037246855990832458995, −1.19736713655978376061576144423, −1.19266747360593074887645854702, −0.979205204091522254618747564705, 0.979205204091522254618747564705, 1.19266747360593074887645854702, 1.19736713655978376061576144423, 1.21023025037246855990832458995, 1.33468046563298565496915827435, 1.40221953573734773365907850071, 1.53848504184469208789448386843, 2.28163051032638861212547266899, 2.42679756532740297964386241743, 2.43469467811386696489074722890, 2.85014966305997503179814267282, 3.07571440787870853012824560642, 3.30902955590238593155924983351, 3.39222756409331222170147485735, 3.49659974432721453271525068978, 3.55107276389428130898270893044, 3.59538630973157184628621057825, 3.94222125156778967567481107005, 4.03793193164359412052108765761, 4.17988633345188948066322439203, 4.31638421287605068385708901979, 4.37970940487822485603434622975, 4.60992542254289008057490805557, 4.61706280632551662549072666168, 4.74310753747249420253072960649
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.