Dirichlet series
| L(s) = 1 | − 7·2-s − 7·3-s + 28·4-s + 2·5-s + 49·6-s − 4·7-s − 84·8-s + 28·9-s − 14·10-s + 7·11-s − 196·12-s − 7·13-s + 28·14-s − 14·15-s + 210·16-s − 4·17-s − 196·18-s − 4·19-s + 56·20-s + 28·21-s − 49·22-s − 23-s + 588·24-s − 13·25-s + 49·26-s − 84·27-s − 112·28-s + ⋯ |
| L(s) = 1 | − 4.94·2-s − 4.04·3-s + 14·4-s + 0.894·5-s + 20.0·6-s − 1.51·7-s − 29.6·8-s + 28/3·9-s − 4.42·10-s + 2.11·11-s − 56.5·12-s − 1.94·13-s + 7.48·14-s − 3.61·15-s + 52.5·16-s − 0.970·17-s − 46.1·18-s − 0.917·19-s + 12.5·20-s + 6.11·21-s − 10.4·22-s − 0.208·23-s + 120.·24-s − 2.59·25-s + 9.60·26-s − 16.1·27-s − 21.1·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{7} \cdot 11^{7} \cdot 61^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{7} \cdot 11^{7} \cdot 61^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{7} \cdot 3^{7} \cdot 11^{7} \cdot 61^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.54859\times 10^{10}\) |
| Root analytic conductor: | \(5.66990\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{7} \cdot 3^{7} \cdot 11^{7} \cdot 61^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.04572829533\) |
| \(L(\frac12)\) | \(\approx\) | \(0.04572829533\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{7} \) |
| 3 | \( ( 1 + T )^{7} \) | |
| 11 | \( ( 1 - T )^{7} \) | |
| 61 | \( ( 1 + T )^{7} \) | |
| good | 5 | \( 1 - 2 T + 17 T^{2} - 7 p T^{3} + 179 T^{4} - 307 T^{5} + 248 p T^{6} - 1902 T^{7} + 248 p^{2} T^{8} - 307 p^{2} T^{9} + 179 p^{3} T^{10} - 7 p^{5} T^{11} + 17 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 + 4 T + 4 p T^{2} + 9 p T^{3} + 258 T^{4} + 284 T^{5} + 1327 T^{6} + 674 T^{7} + 1327 p T^{8} + 284 p^{2} T^{9} + 258 p^{3} T^{10} + 9 p^{5} T^{11} + 4 p^{6} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 7 T + 64 T^{2} + 304 T^{3} + 1602 T^{4} + 5726 T^{5} + 1825 p T^{6} + 76838 T^{7} + 1825 p^{2} T^{8} + 5726 p^{2} T^{9} + 1602 p^{3} T^{10} + 304 p^{4} T^{11} + 64 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 4 T + 27 T^{2} + 62 T^{3} + 940 T^{4} + 3746 T^{5} + 18340 T^{6} + 38608 T^{7} + 18340 p T^{8} + 3746 p^{2} T^{9} + 940 p^{3} T^{10} + 62 p^{4} T^{11} + 27 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 4 T + 46 T^{2} + 167 T^{3} + 1308 T^{4} + 4592 T^{5} + 30835 T^{6} + 110842 T^{7} + 30835 p T^{8} + 4592 p^{2} T^{9} + 1308 p^{3} T^{10} + 167 p^{4} T^{11} + 46 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + T + 54 T^{2} + 2 T^{3} + 1067 T^{4} - 5429 T^{5} + 5978 T^{6} - 217660 T^{7} + 5978 p T^{8} - 5429 p^{2} T^{9} + 1067 p^{3} T^{10} + 2 p^{4} T^{11} + 54 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 6 T + 147 T^{2} - 583 T^{3} + 9622 T^{4} - 30403 T^{5} + 413142 T^{6} - 1095792 T^{7} + 413142 p T^{8} - 30403 p^{2} T^{9} + 9622 p^{3} T^{10} - 583 p^{4} T^{11} + 147 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 7 T + 140 T^{2} - 998 T^{3} + 10018 T^{4} - 66484 T^{5} + 455561 T^{6} - 2619926 T^{7} + 455561 p T^{8} - 66484 p^{2} T^{9} + 10018 p^{3} T^{10} - 998 p^{4} T^{11} + 140 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 15 T + 276 T^{2} + 2934 T^{3} + 31287 T^{4} + 251475 T^{5} + 1939820 T^{6} + 12092656 T^{7} + 1939820 p T^{8} + 251475 p^{2} T^{9} + 31287 p^{3} T^{10} + 2934 p^{4} T^{11} + 276 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - T + 169 T^{2} + 8 T^{3} + 15440 T^{4} + 3400 T^{5} + 918302 T^{6} + 260626 T^{7} + 918302 p T^{8} + 3400 p^{2} T^{9} + 15440 p^{3} T^{10} + 8 p^{4} T^{11} + 169 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 13 T + 222 T^{2} + 1874 T^{3} + 19311 T^{4} + 123943 T^{5} + 1041534 T^{6} + 5818180 T^{7} + 1041534 p T^{8} + 123943 p^{2} T^{9} + 19311 p^{3} T^{10} + 1874 p^{4} T^{11} + 222 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 11 T + 293 T^{2} - 2333 T^{3} + 36491 T^{4} - 231353 T^{5} + 2683487 T^{6} - 13771198 T^{7} + 2683487 p T^{8} - 231353 p^{2} T^{9} + 36491 p^{3} T^{10} - 2333 p^{4} T^{11} + 293 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 14 T + 306 T^{2} - 2995 T^{3} + 39352 T^{4} - 306178 T^{5} + 3066361 T^{6} - 19775306 T^{7} + 3066361 p T^{8} - 306178 p^{2} T^{9} + 39352 p^{3} T^{10} - 2995 p^{4} T^{11} + 306 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 39 T + 922 T^{2} - 15604 T^{3} + 210106 T^{4} - 2343710 T^{5} + 22337347 T^{6} - 183953230 T^{7} + 22337347 p T^{8} - 2343710 p^{2} T^{9} + 210106 p^{3} T^{10} - 15604 p^{4} T^{11} + 922 p^{5} T^{12} - 39 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 3 T + 137 T^{2} + 123 T^{3} + 11105 T^{4} - 43679 T^{5} + 545553 T^{6} - 4043198 T^{7} + 545553 p T^{8} - 43679 p^{2} T^{9} + 11105 p^{3} T^{10} + 123 p^{4} T^{11} + 137 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 12 T + 218 T^{2} - 1673 T^{3} + 28749 T^{4} - 224564 T^{5} + 2781012 T^{6} - 17115998 T^{7} + 2781012 p T^{8} - 224564 p^{2} T^{9} + 28749 p^{3} T^{10} - 1673 p^{4} T^{11} + 218 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 21 T + 570 T^{2} + 8142 T^{3} + 128073 T^{4} + 1380513 T^{5} + 15660896 T^{6} + 131172464 T^{7} + 15660896 p T^{8} + 1380513 p^{2} T^{9} + 128073 p^{3} T^{10} + 8142 p^{4} T^{11} + 570 p^{5} T^{12} + 21 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 15 T + 479 T^{2} - 5323 T^{3} + 99247 T^{4} - 874109 T^{5} + 12066833 T^{6} - 86316370 T^{7} + 12066833 p T^{8} - 874109 p^{2} T^{9} + 99247 p^{3} T^{10} - 5323 p^{4} T^{11} + 479 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 5 T + 376 T^{2} - 1556 T^{3} + 70427 T^{4} - 248779 T^{5} + 8501768 T^{6} - 25379752 T^{7} + 8501768 p T^{8} - 248779 p^{2} T^{9} + 70427 p^{3} T^{10} - 1556 p^{4} T^{11} + 376 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 8 T + 433 T^{2} + 3367 T^{3} + 90341 T^{4} + 655427 T^{5} + 11810964 T^{6} + 74474014 T^{7} + 11810964 p T^{8} + 655427 p^{2} T^{9} + 90341 p^{3} T^{10} + 3367 p^{4} T^{11} + 433 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 20 T + 677 T^{2} + 10031 T^{3} + 194235 T^{4} + 2229699 T^{5} + 31031912 T^{6} + 280309906 T^{7} + 31031912 p T^{8} + 2229699 p^{2} T^{9} + 194235 p^{3} T^{10} + 10031 p^{4} T^{11} + 677 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.84738195692516192801970323602, −3.70941387882118597546688903274, −3.69281093455108603487321660543, −3.63530843715426818482022183992, −3.11108974455509434064019981662, −3.07375378136506432264168546872, −2.86768324499019622930677313832, −2.70892179408267092418070114097, −2.68529044138012132588155103306, −2.65762202794056096400243566574, −2.29067844632534038366763734271, −1.97580597138797820330126330594, −1.94696014558382199391399370927, −1.89490564368268860193122656658, −1.73910424334745096294938686931, −1.70144266318788052513335336424, −1.68167050462008812849459547992, −1.56035326036788537971335949864, −0.940719300886972937553151403899, −0.922462178476051863409495181789, −0.65482578521508102995810035197, −0.64878261200200240676386139257, −0.56053266485974620319476504499, −0.37774563507719881413310557748, −0.15288877524271818723834015230, 0.15288877524271818723834015230, 0.37774563507719881413310557748, 0.56053266485974620319476504499, 0.64878261200200240676386139257, 0.65482578521508102995810035197, 0.922462178476051863409495181789, 0.940719300886972937553151403899, 1.56035326036788537971335949864, 1.68167050462008812849459547992, 1.70144266318788052513335336424, 1.73910424334745096294938686931, 1.89490564368268860193122656658, 1.94696014558382199391399370927, 1.97580597138797820330126330594, 2.29067844632534038366763734271, 2.65762202794056096400243566574, 2.68529044138012132588155103306, 2.70892179408267092418070114097, 2.86768324499019622930677313832, 3.07375378136506432264168546872, 3.11108974455509434064019981662, 3.63530843715426818482022183992, 3.69281093455108603487321660543, 3.70941387882118597546688903274, 3.84738195692516192801970323602
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.