Dirichlet series
| L(s) = 1 | + 2-s + 4·4-s − 3·5-s + 2·7-s + 2·8-s − 3·10-s − 5·11-s + 5·13-s + 2·14-s + 9·16-s + 11·17-s − 9·19-s − 12·20-s − 5·22-s + 16·23-s + 12·25-s + 5·26-s + 8·28-s + 9·29-s − 11·31-s + 14·32-s + 11·34-s − 6·35-s + 6·37-s − 9·38-s − 6·40-s + 22·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 2·4-s − 1.34·5-s + 0.755·7-s + 0.707·8-s − 0.948·10-s − 1.50·11-s + 1.38·13-s + 0.534·14-s + 9/4·16-s + 2.66·17-s − 2.06·19-s − 2.68·20-s − 1.06·22-s + 3.33·23-s + 12/5·25-s + 0.980·26-s + 1.51·28-s + 1.67·29-s − 1.97·31-s + 2.47·32-s + 1.88·34-s − 1.01·35-s + 0.986·37-s − 1.45·38-s − 0.948·40-s + 3.43·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{16} \cdot 7^{8} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(879189.\) |
| Root analytic conductor: | \(2.35236\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{16} \cdot 7^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(22.28308934\) |
| \(L(\frac12)\) | \(\approx\) | \(22.28308934\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) | |
| 11 | \( 1 + 5 T + 26 T^{2} + 75 T^{3} + 251 T^{4} + 75 p T^{5} + 26 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 2 | \( 1 - T - 3 T^{2} + 5 T^{3} - 19 T^{5} + 21 T^{6} + 5 p^{2} T^{7} - 51 T^{8} + 5 p^{3} T^{9} + 21 p^{2} T^{10} - 19 p^{3} T^{11} + 5 p^{5} T^{13} - 3 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 + 3 T - 3 T^{2} - 6 p T^{3} - 39 T^{4} + 6 p^{2} T^{5} + 428 T^{6} - 231 T^{7} - 2169 T^{8} - 231 p T^{9} + 428 p^{2} T^{10} + 6 p^{5} T^{11} - 39 p^{4} T^{12} - 6 p^{6} T^{13} - 3 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 5 T + 35 T^{2} - 90 T^{3} + 47 p T^{4} - 670 T^{5} + 4380 T^{6} + 8865 T^{7} + 38151 T^{8} + 8865 p T^{9} + 4380 p^{2} T^{10} - 670 p^{3} T^{11} + 47 p^{5} T^{12} - 90 p^{5} T^{13} + 35 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 11 T + 39 T^{2} - 139 T^{3} + 1249 T^{4} - 7484 T^{5} + 34098 T^{6} - 128922 T^{7} + 457683 T^{8} - 128922 p T^{9} + 34098 p^{2} T^{10} - 7484 p^{3} T^{11} + 1249 p^{4} T^{12} - 139 p^{5} T^{13} + 39 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 9 T + 43 T^{2} + 9 p T^{3} + 1023 T^{4} + 1674 T^{5} - 12940 T^{6} - 86580 T^{7} - 253639 T^{8} - 86580 p T^{9} - 12940 p^{2} T^{10} + 1674 p^{3} T^{11} + 1023 p^{4} T^{12} + 9 p^{6} T^{13} + 43 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 - 8 T + 83 T^{2} - 402 T^{3} + 2555 T^{4} - 402 p T^{5} + 83 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 - 9 T - 22 T^{2} + 429 T^{3} - 762 T^{4} - 11754 T^{5} + 81940 T^{6} + 191700 T^{7} - 3727369 T^{8} + 191700 p T^{9} + 81940 p^{2} T^{10} - 11754 p^{3} T^{11} - 762 p^{4} T^{12} + 429 p^{5} T^{13} - 22 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 11 T + 34 T^{2} + 183 T^{3} + 3210 T^{4} + 12836 T^{5} - 31430 T^{6} + 25700 T^{7} + 2517721 T^{8} + 25700 p T^{9} - 31430 p^{2} T^{10} + 12836 p^{3} T^{11} + 3210 p^{4} T^{12} + 183 p^{5} T^{13} + 34 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 6 T - 63 T^{2} + 520 T^{3} + 1350 T^{4} - 12954 T^{5} - 20209 T^{6} + 92390 T^{7} + 1366879 T^{8} + 92390 p T^{9} - 20209 p^{2} T^{10} - 12954 p^{3} T^{11} + 1350 p^{4} T^{12} + 520 p^{5} T^{13} - 63 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 22 T + 160 T^{2} - 414 T^{3} + 2641 T^{4} - 46342 T^{5} + 334288 T^{6} - 25916 p T^{7} + 2274033 T^{8} - 25916 p^{2} T^{9} + 334288 p^{2} T^{10} - 46342 p^{3} T^{11} + 2641 p^{4} T^{12} - 414 p^{5} T^{13} + 160 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) | |
| 47 | \( 1 + 7 T - 100 T^{2} - 1414 T^{3} + 286 T^{4} + 103523 T^{5} + 579636 T^{6} - 2543800 T^{7} - 42803781 T^{8} - 2543800 p T^{9} + 579636 p^{2} T^{10} + 103523 p^{3} T^{11} + 286 p^{4} T^{12} - 1414 p^{5} T^{13} - 100 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 2 T - 173 T^{2} - 964 T^{3} + 9936 T^{4} + 104742 T^{5} + 122655 T^{6} - 3229712 T^{7} - 31796061 T^{8} - 3229712 p T^{9} + 122655 p^{2} T^{10} + 104742 p^{3} T^{11} + 9936 p^{4} T^{12} - 964 p^{5} T^{13} - 173 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 25 T + 206 T^{2} - 25 T^{3} - 15220 T^{4} - 157300 T^{5} - 429276 T^{6} + 8492650 T^{7} + 113408229 T^{8} + 8492650 p T^{9} - 429276 p^{2} T^{10} - 157300 p^{3} T^{11} - 15220 p^{4} T^{12} - 25 p^{5} T^{13} + 206 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 7 T - 108 T^{2} + 1783 T^{3} + 318 T^{4} - 164342 T^{5} + 1092660 T^{6} + 5279480 T^{7} - 100824709 T^{8} + 5279480 p T^{9} + 1092660 p^{2} T^{10} - 164342 p^{3} T^{11} + 318 p^{4} T^{12} + 1783 p^{5} T^{13} - 108 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + 15 T + 5 p T^{2} + 3060 T^{3} + 35713 T^{4} + 3060 p T^{5} + 5 p^{3} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 - 14 T - 9 T^{2} + 147 T^{3} + 6927 T^{4} - 52059 T^{5} + 728307 T^{6} - 4047848 T^{7} - 14238189 T^{8} - 4047848 p T^{9} + 728307 p^{2} T^{10} - 52059 p^{3} T^{11} + 6927 p^{4} T^{12} + 147 p^{5} T^{13} - 9 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 3 T - 2 T^{2} - 345 T^{3} + 220 T^{4} - 100292 T^{5} + 620974 T^{6} + 13580 T^{7} + 31898639 T^{8} + 13580 p T^{9} + 620974 p^{2} T^{10} - 100292 p^{3} T^{11} + 220 p^{4} T^{12} - 345 p^{5} T^{13} - 2 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 9 T - 75 T^{2} - 768 T^{3} + 10031 T^{4} + 24564 T^{5} - 1553178 T^{6} - 1473567 T^{7} + 120297853 T^{8} - 1473567 p T^{9} - 1553178 p^{2} T^{10} + 24564 p^{3} T^{11} + 10031 p^{4} T^{12} - 768 p^{5} T^{13} - 75 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 23 T + 98 T^{2} - 1745 T^{3} - 15470 T^{4} + 98632 T^{5} + 1750974 T^{6} - 1504700 T^{7} - 137136651 T^{8} - 1504700 p T^{9} + 1750974 p^{2} T^{10} + 98632 p^{3} T^{11} - 15470 p^{4} T^{12} - 1745 p^{5} T^{13} + 98 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 - 17 T + 312 T^{2} - 3419 T^{3} + 38939 T^{4} - 3419 p T^{5} + 312 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 30 T + 361 T^{2} - 2160 T^{3} + 10062 T^{4} - 83370 T^{5} - 577417 T^{6} + 32754150 T^{7} - 458148745 T^{8} + 32754150 p T^{9} - 577417 p^{2} T^{10} - 83370 p^{3} T^{11} + 10062 p^{4} T^{12} - 2160 p^{5} T^{13} + 361 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.55213339203988183233421395176, −4.38409746298775644446885098699, −4.38249352175883858580943537710, −4.18241997994324040707537979975, −4.01375504898033011078168305076, −3.71409200889205907212782224593, −3.66204220486828299916698981816, −3.43196522274180202200582549876, −3.42056539807196959003015278642, −3.34012362545872837336299739906, −3.11323989260059587960993426695, −2.90224228555753674183278225840, −2.77974087146392383975292270184, −2.73155754064788278777851565508, −2.68108531054094411692549867000, −2.53480934452181999891785586063, −2.19281909157309866997415209883, −2.06976273724158154702959789911, −1.84059808051410924706269074777, −1.50771538840841610261496673702, −1.29684674837659620787922586149, −1.18382856837166344769334047592, −0.864739164790193686678107809974, −0.72533052911107032105685989721, −0.61112569032517841190909038833, 0.61112569032517841190909038833, 0.72533052911107032105685989721, 0.864739164790193686678107809974, 1.18382856837166344769334047592, 1.29684674837659620787922586149, 1.50771538840841610261496673702, 1.84059808051410924706269074777, 2.06976273724158154702959789911, 2.19281909157309866997415209883, 2.53480934452181999891785586063, 2.68108531054094411692549867000, 2.73155754064788278777851565508, 2.77974087146392383975292270184, 2.90224228555753674183278225840, 3.11323989260059587960993426695, 3.34012362545872837336299739906, 3.42056539807196959003015278642, 3.43196522274180202200582549876, 3.66204220486828299916698981816, 3.71409200889205907212782224593, 4.01375504898033011078168305076, 4.18241997994324040707537979975, 4.38249352175883858580943537710, 4.38409746298775644446885098699, 4.55213339203988183233421395176
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.