Dirichlet series
| L(s) = 1 | − 3·2-s + 5·3-s + 8·4-s + 2·5-s − 15·6-s − 4·7-s − 16·8-s + 18·9-s − 6·10-s + 40·12-s − 3·13-s + 12·14-s + 10·15-s + 29·16-s − 12·17-s − 54·18-s − 5·19-s + 16·20-s − 20·21-s + 10·23-s − 80·24-s + 25-s + 9·26-s + 55·27-s − 32·28-s − 21·29-s − 30·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2.88·3-s + 4·4-s + 0.894·5-s − 6.12·6-s − 1.51·7-s − 5.65·8-s + 6·9-s − 1.89·10-s + 11.5·12-s − 0.832·13-s + 3.20·14-s + 2.58·15-s + 29/4·16-s − 2.91·17-s − 12.7·18-s − 1.14·19-s + 3.57·20-s − 4.36·21-s + 2.08·23-s − 16.3·24-s + 1/5·25-s + 1.76·26-s + 10.5·27-s − 6.04·28-s − 3.89·29-s − 5.47·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(5^{8} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(296660.\) |
| Root analytic conductor: | \(2.19794\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 5^{8} \cdot 11^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.812758314\) |
| \(L(\frac12)\) | \(\approx\) | \(1.812758314\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + 3 T + T^{2} - 5 T^{3} - p^{2} T^{4} + 9 T^{5} + 15 T^{6} - 21 T^{7} - 73 T^{8} - 21 p T^{9} + 15 p^{2} T^{10} + 9 p^{3} T^{11} - p^{6} T^{12} - 5 p^{5} T^{13} + p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 - 5 T + 7 T^{2} + 5 T^{4} - 20 p T^{5} + 142 T^{6} - 25 p T^{7} - 131 T^{8} - 25 p^{2} T^{9} + 142 p^{2} T^{10} - 20 p^{4} T^{11} + 5 p^{4} T^{12} + 7 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 + 4 T + 3 T^{2} - p T^{3} + T^{4} + 239 T^{5} + 635 T^{6} - 326 T^{7} - 3001 T^{8} - 326 p T^{9} + 635 p^{2} T^{10} + 239 p^{3} T^{11} + p^{4} T^{12} - p^{6} T^{13} + 3 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 3 T - 15 T^{2} - 53 T^{3} - 21 T^{4} + 514 T^{5} + 3544 T^{6} - 3906 T^{7} - 69209 T^{8} - 3906 p T^{9} + 3544 p^{2} T^{10} + 514 p^{3} T^{11} - 21 p^{4} T^{12} - 53 p^{5} T^{13} - 15 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 12 T + 47 T^{2} + 154 T^{3} + 1418 T^{4} + 8246 T^{5} + 30029 T^{6} + 131742 T^{7} + 626063 T^{8} + 131742 p T^{9} + 30029 p^{2} T^{10} + 8246 p^{3} T^{11} + 1418 p^{4} T^{12} + 154 p^{5} T^{13} + 47 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 5 T + 22 T^{2} + 98 T^{4} + 1205 T^{5} + 1664 T^{6} - 16450 T^{7} - 239345 T^{8} - 16450 p T^{9} + 1664 p^{2} T^{10} + 1205 p^{3} T^{11} + 98 p^{4} T^{12} + 22 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 - 5 T + 96 T^{2} - 335 T^{3} + 3347 T^{4} - 335 p T^{5} + 96 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 + 21 T + 156 T^{2} + 330 T^{3} - 2322 T^{4} - 25875 T^{5} - 128176 T^{6} - 27564 T^{7} + 2442723 T^{8} - 27564 p T^{9} - 128176 p^{2} T^{10} - 25875 p^{3} T^{11} - 2322 p^{4} T^{12} + 330 p^{5} T^{13} + 156 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 15 T + 63 T^{2} + 245 T^{3} - 2367 T^{4} - 5040 T^{5} + 71336 T^{6} + 370000 T^{7} - 5341895 T^{8} + 370000 p T^{9} + 71336 p^{2} T^{10} - 5040 p^{3} T^{11} - 2367 p^{4} T^{12} + 245 p^{5} T^{13} + 63 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 31 T + 460 T^{2} + 4331 T^{3} + 27314 T^{4} + 88288 T^{5} - 368284 T^{6} - 7742998 T^{7} - 61104929 T^{8} - 7742998 p T^{9} - 368284 p^{2} T^{10} + 88288 p^{3} T^{11} + 27314 p^{4} T^{12} + 4331 p^{5} T^{13} + 460 p^{6} T^{14} + 31 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 3 T - 84 T^{2} - 643 T^{3} + 1026 T^{4} + 36274 T^{5} + 206324 T^{6} - 701364 T^{7} - 13301713 T^{8} - 701364 p T^{9} + 206324 p^{2} T^{10} + 36274 p^{3} T^{11} + 1026 p^{4} T^{12} - 643 p^{5} T^{13} - 84 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 - 19 T + 293 T^{2} - 2740 T^{3} + 21711 T^{4} - 2740 p T^{5} + 293 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 + 5 T - 71 T^{2} - 405 T^{3} + 2887 T^{4} + 16310 T^{5} - 80948 T^{6} - 411190 T^{7} + 132525 T^{8} - 411190 p T^{9} - 80948 p^{2} T^{10} + 16310 p^{3} T^{11} + 2887 p^{4} T^{12} - 405 p^{5} T^{13} - 71 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 2 T - T^{2} - 161 T^{3} + 161 T^{4} - 32839 T^{5} + 67085 T^{6} + 83390 T^{7} + 7457057 T^{8} + 83390 p T^{9} + 67085 p^{2} T^{10} - 32839 p^{3} T^{11} + 161 p^{4} T^{12} - 161 p^{5} T^{13} - p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 18 T + 137 T^{2} - 1107 T^{3} + 11427 T^{4} - 92115 T^{5} + 779923 T^{6} - 6951636 T^{7} + 54843695 T^{8} - 6951636 p T^{9} + 779923 p^{2} T^{10} - 92115 p^{3} T^{11} + 11427 p^{4} T^{12} - 1107 p^{5} T^{13} + 137 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 6 T - 47 T^{2} - 414 T^{3} + 4637 T^{4} + 11250 T^{5} - 439933 T^{6} + 581958 T^{7} + 38065660 T^{8} + 581958 p T^{9} - 439933 p^{2} T^{10} + 11250 p^{3} T^{11} + 4637 p^{4} T^{12} - 414 p^{5} T^{13} - 47 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + 19 T + 290 T^{2} + 2805 T^{3} + 25803 T^{4} + 2805 p T^{5} + 290 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 - 15 T + 16 T^{2} - 165 T^{3} + 14850 T^{4} - 104730 T^{5} + 78254 T^{6} - 4128630 T^{7} + 77353769 T^{8} - 4128630 p T^{9} + 78254 p^{2} T^{10} - 104730 p^{3} T^{11} + 14850 p^{4} T^{12} - 165 p^{5} T^{13} + 16 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 2 T - 65 T^{2} - 298 T^{3} + 10994 T^{4} - 6376 T^{5} - 885361 T^{6} + 818854 T^{7} + 81544771 T^{8} + 818854 p T^{9} - 885361 p^{2} T^{10} - 6376 p^{3} T^{11} + 10994 p^{4} T^{12} - 298 p^{5} T^{13} - 65 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 3 T + 75 T^{2} - 1060 T^{3} + 7725 T^{4} + 44016 T^{5} + 479822 T^{6} + 3127825 T^{7} - 55525715 T^{8} + 3127825 p T^{9} + 479822 p^{2} T^{10} + 44016 p^{3} T^{11} + 7725 p^{4} T^{12} - 1060 p^{5} T^{13} + 75 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 38 T + 587 T^{2} - 5179 T^{3} + 35051 T^{4} - 121633 T^{5} - 1577445 T^{6} + 23545528 T^{7} - 185766921 T^{8} + 23545528 p T^{9} - 1577445 p^{2} T^{10} - 121633 p^{3} T^{11} + 35051 p^{4} T^{12} - 5179 p^{5} T^{13} + 587 p^{6} T^{14} - 38 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 8 T + 254 T^{2} + 1664 T^{3} + 31231 T^{4} + 1664 p T^{5} + 254 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 + 56 T + 1215 T^{2} + 9736 T^{3} - 78156 T^{4} - 2414432 T^{5} - 15563379 T^{6} + 134047232 T^{7} + 2933576351 T^{8} + 134047232 p T^{9} - 15563379 p^{2} T^{10} - 2414432 p^{3} T^{11} - 78156 p^{4} T^{12} + 9736 p^{5} T^{13} + 1215 p^{6} T^{14} + 56 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.46850743641937402148457602038, −4.30233020353358873128462496001, −4.26955004033676091953034830091, −4.26039784723613228593367360503, −4.03194684026880505042893605531, −3.88307999277936950770045276328, −3.68644025371325613403267508495, −3.48719627724127720599627646063, −3.40595830989253007098971557085, −3.14060590749210100741503158183, −3.08420508643665548196841873983, −3.02290619402642691214279821999, −2.84336782846742766736340212573, −2.63812723761394884307158217394, −2.30892045818043559697041177471, −2.24111826040546594877865454939, −2.16842133543638188016739592337, −2.10759082469302385646708992405, −2.09612114075353792966860797197, −1.69736431063964018481726433243, −1.67149842358766245229019965562, −1.21740400565484983719924901005, −1.08286063106032696167277620798, −0.76209708290372485518645155124, −0.18195150161466740869011237138, 0.18195150161466740869011237138, 0.76209708290372485518645155124, 1.08286063106032696167277620798, 1.21740400565484983719924901005, 1.67149842358766245229019965562, 1.69736431063964018481726433243, 2.09612114075353792966860797197, 2.10759082469302385646708992405, 2.16842133543638188016739592337, 2.24111826040546594877865454939, 2.30892045818043559697041177471, 2.63812723761394884307158217394, 2.84336782846742766736340212573, 3.02290619402642691214279821999, 3.08420508643665548196841873983, 3.14060590749210100741503158183, 3.40595830989253007098971557085, 3.48719627724127720599627646063, 3.68644025371325613403267508495, 3.88307999277936950770045276328, 4.03194684026880505042893605531, 4.26039784723613228593367360503, 4.26955004033676091953034830091, 4.30233020353358873128462496001, 4.46850743641937402148457602038
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.