Dirichlet series
| L(s) = 1 | + 2-s − 2·3-s − 2·4-s + 11·5-s − 2·6-s − 5·7-s − 8-s − 2·9-s + 11·10-s + 2·11-s + 4·12-s + 6·13-s − 5·14-s − 22·15-s + 5·17-s − 2·18-s − 10·19-s − 22·20-s + 10·21-s + 2·22-s − 23-s + 2·24-s + 50·25-s + 6·26-s + 5·27-s + 10·28-s + 30·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 4-s + 4.91·5-s − 0.816·6-s − 1.88·7-s − 0.353·8-s − 2/3·9-s + 3.47·10-s + 0.603·11-s + 1.15·12-s + 1.66·13-s − 1.33·14-s − 5.68·15-s + 1.21·17-s − 0.471·18-s − 2.29·19-s − 4.91·20-s + 2.18·21-s + 0.426·22-s − 0.208·23-s + 0.408·24-s + 10·25-s + 1.17·26-s + 0.962·27-s + 1.88·28-s + 5.57·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(139^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(139^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(139^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.07512\) |
| Root analytic conductor: | \(1.05352\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 139^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.882023430\) |
| \(L(\frac12)\) | \(\approx\) | \(1.882023430\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 139 | \( ( 1 - T )^{7} \) |
| good | 2 | \( 1 - T + 3 T^{2} - p^{2} T^{3} + 9 T^{4} - 3 p T^{5} + 9 p T^{6} - p^{4} T^{7} + 9 p^{2} T^{8} - 3 p^{3} T^{9} + 9 p^{3} T^{10} - p^{6} T^{11} + 3 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 + 2 T + 2 p T^{2} + 11 T^{3} + 20 T^{4} + 22 T^{5} + 43 T^{6} + 26 T^{7} + 43 p T^{8} + 22 p^{2} T^{9} + 20 p^{3} T^{10} + 11 p^{4} T^{11} + 2 p^{6} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 5 | \( 1 - 11 T + 71 T^{2} - 328 T^{3} + 1214 T^{4} - 3766 T^{5} + 2031 p T^{6} - 24093 T^{7} + 2031 p^{2} T^{8} - 3766 p^{2} T^{9} + 1214 p^{3} T^{10} - 328 p^{4} T^{11} + 71 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 + 5 T + 41 T^{2} + 128 T^{3} + 594 T^{4} + 1270 T^{5} + 4799 T^{6} + 8663 T^{7} + 4799 p T^{8} + 1270 p^{2} T^{9} + 594 p^{3} T^{10} + 128 p^{4} T^{11} + 41 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 2 T + 41 T^{2} - 50 T^{3} + 747 T^{4} - 336 T^{5} + 8869 T^{6} - 387 T^{7} + 8869 p T^{8} - 336 p^{2} T^{9} + 747 p^{3} T^{10} - 50 p^{4} T^{11} + 41 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 6 T + 89 T^{2} - 404 T^{3} + 3311 T^{4} - 11844 T^{5} + 69309 T^{6} - 197757 T^{7} + 69309 p T^{8} - 11844 p^{2} T^{9} + 3311 p^{3} T^{10} - 404 p^{4} T^{11} + 89 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 5 T + 77 T^{2} - 147 T^{3} + 1585 T^{4} + 3829 T^{5} + 3881 T^{6} + 165878 T^{7} + 3881 p T^{8} + 3829 p^{2} T^{9} + 1585 p^{3} T^{10} - 147 p^{4} T^{11} + 77 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 10 T + 130 T^{2} + 927 T^{3} + 7094 T^{4} + 39234 T^{5} + 218745 T^{6} + 50334 p T^{7} + 218745 p T^{8} + 39234 p^{2} T^{9} + 7094 p^{3} T^{10} + 927 p^{4} T^{11} + 130 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + T + 113 T^{2} + 3 T^{3} + 5837 T^{4} - 3577 T^{5} + 189029 T^{6} - 144326 T^{7} + 189029 p T^{8} - 3577 p^{2} T^{9} + 5837 p^{3} T^{10} + 3 p^{4} T^{11} + 113 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 30 T + 503 T^{2} - 5736 T^{3} + 49929 T^{4} - 12142 p T^{5} + 2152887 T^{6} - 11980823 T^{7} + 2152887 p T^{8} - 12142 p^{3} T^{9} + 49929 p^{3} T^{10} - 5736 p^{4} T^{11} + 503 p^{5} T^{12} - 30 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 20 T + 313 T^{2} + 3540 T^{3} + 33819 T^{4} + 267438 T^{5} + 1851523 T^{6} + 10966915 T^{7} + 1851523 p T^{8} + 267438 p^{2} T^{9} + 33819 p^{3} T^{10} + 3540 p^{4} T^{11} + 313 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 6 T + 103 T^{2} - 897 T^{3} + 7857 T^{4} - 56685 T^{5} + 420206 T^{6} - 2498246 T^{7} + 420206 p T^{8} - 56685 p^{2} T^{9} + 7857 p^{3} T^{10} - 897 p^{4} T^{11} + 103 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 19 T + 184 T^{2} - 1087 T^{3} + 4 p^{2} T^{4} - 1413 p T^{5} + 542627 T^{6} - 3906482 T^{7} + 542627 p T^{8} - 1413 p^{3} T^{9} + 4 p^{5} T^{10} - 1087 p^{4} T^{11} + 184 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 12 T + 246 T^{2} + 1651 T^{3} + 18912 T^{4} + 65268 T^{5} + 704463 T^{6} + 1413290 T^{7} + 704463 p T^{8} + 65268 p^{2} T^{9} + 18912 p^{3} T^{10} + 1651 p^{4} T^{11} + 246 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 3 T + 109 T^{2} - 37 T^{3} + 9701 T^{4} + 5669 T^{5} + 602068 T^{6} - 199798 T^{7} + 602068 p T^{8} + 5669 p^{2} T^{9} + 9701 p^{3} T^{10} - 37 p^{4} T^{11} + 109 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 38 T + 918 T^{2} - 15753 T^{3} + 214716 T^{4} - 2385562 T^{5} + 22273641 T^{6} - 175687038 T^{7} + 22273641 p T^{8} - 2385562 p^{2} T^{9} + 214716 p^{3} T^{10} - 15753 p^{4} T^{11} + 918 p^{5} T^{12} - 38 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 14 T + 358 T^{2} + 4141 T^{3} + 58224 T^{4} + 550458 T^{5} + 5494079 T^{6} + 41878902 T^{7} + 5494079 p T^{8} + 550458 p^{2} T^{9} + 58224 p^{3} T^{10} + 4141 p^{4} T^{11} + 358 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 4 T + 168 T^{2} - 931 T^{3} + 16996 T^{4} - 101432 T^{5} + 1438575 T^{6} - 7223874 T^{7} + 1438575 p T^{8} - 101432 p^{2} T^{9} + 16996 p^{3} T^{10} - 931 p^{4} T^{11} + 168 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 9 T + 252 T^{2} - 2212 T^{3} + 36841 T^{4} - 280719 T^{5} + 3525326 T^{6} - 23240880 T^{7} + 3525326 p T^{8} - 280719 p^{2} T^{9} + 36841 p^{3} T^{10} - 2212 p^{4} T^{11} + 252 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 24 T + 531 T^{2} - 7902 T^{3} + 110959 T^{4} - 1236210 T^{5} + 12915763 T^{6} - 111985073 T^{7} + 12915763 p T^{8} - 1236210 p^{2} T^{9} + 110959 p^{3} T^{10} - 7902 p^{4} T^{11} + 531 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 5 T + 241 T^{2} + 1463 T^{3} + 29835 T^{4} + 233795 T^{5} + 2631931 T^{6} + 21987634 T^{7} + 2631931 p T^{8} + 233795 p^{2} T^{9} + 29835 p^{3} T^{10} + 1463 p^{4} T^{11} + 241 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 8 T + 291 T^{2} - 2542 T^{3} + 45327 T^{4} - 424734 T^{5} + 4781091 T^{6} - 42061795 T^{7} + 4781091 p T^{8} - 424734 p^{2} T^{9} + 45327 p^{3} T^{10} - 2542 p^{4} T^{11} + 291 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 9 T + 265 T^{2} + 1492 T^{3} + 31458 T^{4} + 105574 T^{5} + 2627287 T^{6} + 6171795 T^{7} + 2627287 p T^{8} + 105574 p^{2} T^{9} + 31458 p^{3} T^{10} + 1492 p^{4} T^{11} + 265 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 10 T + 385 T^{2} - 3364 T^{3} + 76259 T^{4} - 584084 T^{5} + 9889117 T^{6} - 63995331 T^{7} + 9889117 p T^{8} - 584084 p^{2} T^{9} + 76259 p^{3} T^{10} - 3364 p^{4} T^{11} + 385 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 5 T + 513 T^{2} + 1695 T^{3} + 120449 T^{4} + 268555 T^{5} + 17291453 T^{6} + 29069346 T^{7} + 17291453 p T^{8} + 268555 p^{2} T^{9} + 120449 p^{3} T^{10} + 1695 p^{4} T^{11} + 513 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−6.32942821830898514877625774279, −6.31771372530985146828738308465, −6.28236602205375346718529176661, −6.27196928265547203409141944476, −6.06774503100792385421125576329, −5.69533840313073123655199724930, −5.57346371656231028443552594482, −5.44580997586798804423401857281, −5.25608842699554926932562055754, −5.18798717567746075684120896919, −5.14549047835161607733112285825, −4.71885933750437545998620785463, −4.34532943215737786972727556586, −4.17572040624675997311296066934, −4.00411638681954913953595376329, −3.77744209725969877219734300160, −3.62386858026789732882684642320, −3.10305270973767044169088182244, −2.91440291277158682988884182729, −2.68297200654132790243115594372, −2.33264701009223557130826448329, −2.15772150637850850685135139345, −1.75394998814133913255523014847, −1.58075068472232777858525680636, −0.894103374391590948326768822514, 0.894103374391590948326768822514, 1.58075068472232777858525680636, 1.75394998814133913255523014847, 2.15772150637850850685135139345, 2.33264701009223557130826448329, 2.68297200654132790243115594372, 2.91440291277158682988884182729, 3.10305270973767044169088182244, 3.62386858026789732882684642320, 3.77744209725969877219734300160, 4.00411638681954913953595376329, 4.17572040624675997311296066934, 4.34532943215737786972727556586, 4.71885933750437545998620785463, 5.14549047835161607733112285825, 5.18798717567746075684120896919, 5.25608842699554926932562055754, 5.44580997586798804423401857281, 5.57346371656231028443552594482, 5.69533840313073123655199724930, 6.06774503100792385421125576329, 6.27196928265547203409141944476, 6.28236602205375346718529176661, 6.31771372530985146828738308465, 6.32942821830898514877625774279
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.