Dirichlet series
| L(s) = 1 | + 2-s + 8·3-s + 5·5-s + 8·6-s + 2·8-s + 36·9-s + 5·10-s + 10·11-s − 6·13-s + 40·15-s − 2·16-s + 21·17-s + 36·18-s + 5·19-s + 10·22-s − 8·23-s + 16·24-s + 6·25-s − 6·26-s + 120·27-s + 2·29-s + 40·30-s − 13·31-s − 2·32-s + 80·33-s + 21·34-s + 13·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4.61·3-s + 2.23·5-s + 3.26·6-s + 0.707·8-s + 12·9-s + 1.58·10-s + 3.01·11-s − 1.66·13-s + 10.3·15-s − 1/2·16-s + 5.09·17-s + 8.48·18-s + 1.14·19-s + 2.13·22-s − 1.66·23-s + 3.26·24-s + 6/5·25-s − 1.17·26-s + 23.0·27-s + 0.371·29-s + 7.30·30-s − 2.33·31-s − 0.353·32-s + 13.9·33-s + 3.60·34-s + 2.13·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 7^{16} \cdot 23^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.82213\times 10^{11}\) |
| Root analytic conductor: | \(5.19590\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 7^{16} \cdot 23^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1832.662775\) |
| \(L(\frac12)\) | \(\approx\) | \(1832.662775\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{8} \) |
| 7 | \( 1 \) | |
| 23 | \( ( 1 + T )^{8} \) | |
| good | 2 | \( 1 - T + T^{2} - 3 T^{3} + 7 T^{4} - 9 T^{5} + 7 T^{6} - 13 T^{7} + 9 p^{2} T^{8} - 13 p T^{9} + 7 p^{2} T^{10} - 9 p^{3} T^{11} + 7 p^{4} T^{12} - 3 p^{5} T^{13} + p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 - p T + 19 T^{2} - 48 T^{3} + 151 T^{4} - 383 T^{5} + 1036 T^{6} - 2107 T^{7} + 1022 p T^{8} - 2107 p T^{9} + 1036 p^{2} T^{10} - 383 p^{3} T^{11} + 151 p^{4} T^{12} - 48 p^{5} T^{13} + 19 p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 10 T + 84 T^{2} - 514 T^{3} + 2820 T^{4} - 13194 T^{5} + 56604 T^{6} - 214546 T^{7} + 752790 T^{8} - 214546 p T^{9} + 56604 p^{2} T^{10} - 13194 p^{3} T^{11} + 2820 p^{4} T^{12} - 514 p^{5} T^{13} + 84 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 6 T + 66 T^{2} + 388 T^{3} + 2428 T^{4} + 11954 T^{5} + 4340 p T^{6} + 232990 T^{7} + 881265 T^{8} + 232990 p T^{9} + 4340 p^{3} T^{10} + 11954 p^{3} T^{11} + 2428 p^{4} T^{12} + 388 p^{5} T^{13} + 66 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 21 T + 275 T^{2} - 2572 T^{3} + 19559 T^{4} - 124943 T^{5} + 698916 T^{6} - 3436817 T^{7} + 15060332 T^{8} - 3436817 p T^{9} + 698916 p^{2} T^{10} - 124943 p^{3} T^{11} + 19559 p^{4} T^{12} - 2572 p^{5} T^{13} + 275 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 5 T + 58 T^{2} - 191 T^{3} + 1874 T^{4} - 6259 T^{5} + 52034 T^{6} - 149089 T^{7} + 1060158 T^{8} - 149089 p T^{9} + 52034 p^{2} T^{10} - 6259 p^{3} T^{11} + 1874 p^{4} T^{12} - 191 p^{5} T^{13} + 58 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 2 T + 124 T^{2} - 6 p T^{3} + 7492 T^{4} - 6642 T^{5} + 10292 p T^{6} - 160814 T^{7} + 9378198 T^{8} - 160814 p T^{9} + 10292 p^{3} T^{10} - 6642 p^{3} T^{11} + 7492 p^{4} T^{12} - 6 p^{6} T^{13} + 124 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 13 T + 226 T^{2} + 2149 T^{3} + 724 p T^{4} + 166369 T^{5} + 1301710 T^{6} + 7839289 T^{7} + 49198038 T^{8} + 7839289 p T^{9} + 1301710 p^{2} T^{10} + 166369 p^{3} T^{11} + 724 p^{5} T^{12} + 2149 p^{5} T^{13} + 226 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 13 T + 244 T^{2} - 2551 T^{3} + 27964 T^{4} - 236713 T^{5} + 1929868 T^{6} - 13379899 T^{7} + 87195366 T^{8} - 13379899 p T^{9} + 1929868 p^{2} T^{10} - 236713 p^{3} T^{11} + 27964 p^{4} T^{12} - 2551 p^{5} T^{13} + 244 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 16 T + 368 T^{2} - 4320 T^{3} + 56180 T^{4} - 510560 T^{5} + 4749568 T^{6} - 34206912 T^{7} + 245662006 T^{8} - 34206912 p T^{9} + 4749568 p^{2} T^{10} - 510560 p^{3} T^{11} + 56180 p^{4} T^{12} - 4320 p^{5} T^{13} + 368 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 15 T + 260 T^{2} - 2301 T^{3} + 24654 T^{4} - 169237 T^{5} + 1508652 T^{6} - 9048167 T^{7} + 72141822 T^{8} - 9048167 p T^{9} + 1508652 p^{2} T^{10} - 169237 p^{3} T^{11} + 24654 p^{4} T^{12} - 2301 p^{5} T^{13} + 260 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + T + 191 T^{2} - 322 T^{3} + 19135 T^{4} - 55425 T^{5} + 1374358 T^{6} - 4430231 T^{7} + 73261448 T^{8} - 4430231 p T^{9} + 1374358 p^{2} T^{10} - 55425 p^{3} T^{11} + 19135 p^{4} T^{12} - 322 p^{5} T^{13} + 191 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 3 T + 311 T^{2} - 1014 T^{3} + 46369 T^{4} - 147837 T^{5} + 4328194 T^{6} - 12405075 T^{7} + 275137826 T^{8} - 12405075 p T^{9} + 4328194 p^{2} T^{10} - 147837 p^{3} T^{11} + 46369 p^{4} T^{12} - 1014 p^{5} T^{13} + 311 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 26 T + 572 T^{2} - 8446 T^{3} + 112660 T^{4} - 1222134 T^{5} + 12382564 T^{6} - 107981666 T^{7} + 888609686 T^{8} - 107981666 p T^{9} + 12382564 p^{2} T^{10} - 1222134 p^{3} T^{11} + 112660 p^{4} T^{12} - 8446 p^{5} T^{13} + 572 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 14 T + 336 T^{2} + 3898 T^{3} + 55300 T^{4} + 539950 T^{5} + 5808800 T^{6} + 48188970 T^{7} + 422117142 T^{8} + 48188970 p T^{9} + 5808800 p^{2} T^{10} + 539950 p^{3} T^{11} + 55300 p^{4} T^{12} + 3898 p^{5} T^{13} + 336 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 38 T + 940 T^{2} - 16916 T^{3} + 251234 T^{4} - 3155944 T^{5} + 34735964 T^{6} - 336989032 T^{7} + 2921754033 T^{8} - 336989032 p T^{9} + 34735964 p^{2} T^{10} - 3155944 p^{3} T^{11} + 251234 p^{4} T^{12} - 16916 p^{5} T^{13} + 940 p^{6} T^{14} - 38 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 9 T + 147 T^{2} - 18 p T^{3} + 19707 T^{4} - 151303 T^{5} + 1665922 T^{6} - 12066733 T^{7} + 136917320 T^{8} - 12066733 p T^{9} + 1665922 p^{2} T^{10} - 151303 p^{3} T^{11} + 19707 p^{4} T^{12} - 18 p^{6} T^{13} + 147 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 6 T + 330 T^{2} + 2120 T^{3} + 58292 T^{4} + 349062 T^{5} + 6965940 T^{6} + 37007230 T^{7} + 595403013 T^{8} + 37007230 p T^{9} + 6965940 p^{2} T^{10} + 349062 p^{3} T^{11} + 58292 p^{4} T^{12} + 2120 p^{5} T^{13} + 330 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 23 T + 454 T^{2} - 6093 T^{3} + 72762 T^{4} - 717749 T^{5} + 6607318 T^{6} - 55187399 T^{7} + 490701918 T^{8} - 55187399 p T^{9} + 6607318 p^{2} T^{10} - 717749 p^{3} T^{11} + 72762 p^{4} T^{12} - 6093 p^{5} T^{13} + 454 p^{6} T^{14} - 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 30 T + 700 T^{2} - 12606 T^{3} + 197716 T^{4} - 2623518 T^{5} + 31495012 T^{6} - 333591438 T^{7} + 3218957302 T^{8} - 333591438 p T^{9} + 31495012 p^{2} T^{10} - 2623518 p^{3} T^{11} + 197716 p^{4} T^{12} - 12606 p^{5} T^{13} + 700 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 12 T + 72 T^{2} - 202 T^{3} + 12544 T^{4} - 124182 T^{5} + 1409300 T^{6} - 10159792 T^{7} + 139894014 T^{8} - 10159792 p T^{9} + 1409300 p^{2} T^{10} - 124182 p^{3} T^{11} + 12544 p^{4} T^{12} - 202 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 14 T + 476 T^{2} - 5622 T^{3} + 107340 T^{4} - 1072658 T^{5} + 15587876 T^{6} - 135297754 T^{7} + 1697234694 T^{8} - 135297754 p T^{9} + 15587876 p^{2} T^{10} - 1072658 p^{3} T^{11} + 107340 p^{4} T^{12} - 5622 p^{5} T^{13} + 476 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.55190946029908857803985925750, −3.47899886458601509622706444709, −3.37427263367647110572417079281, −3.21635398445787068300970969991, −3.13635093418087975154235134300, −3.11025418437386516967000604337, −2.70118135464532415031873530675, −2.62271092355865513798338354909, −2.47702707615170243842223421271, −2.47089798466739842609899493176, −2.45860614398229720222162856933, −2.29992415055520986592992191184, −2.22089217179805665196473659483, −2.02257070361929152330437636783, −1.98549291678652335157863152148, −1.89832728329609395989901349904, −1.63646277832710497953569223805, −1.46850567810441648637292449420, −1.43456896513354243711243155540, −1.28396740204545717468602456585, −0.960576602974353706778534829437, −0.907483187083727688176828941136, −0.897671532404896029465725142555, −0.65629615524673451559300051357, −0.63620368501603698354334765808, 0.63620368501603698354334765808, 0.65629615524673451559300051357, 0.897671532404896029465725142555, 0.907483187083727688176828941136, 0.960576602974353706778534829437, 1.28396740204545717468602456585, 1.43456896513354243711243155540, 1.46850567810441648637292449420, 1.63646277832710497953569223805, 1.89832728329609395989901349904, 1.98549291678652335157863152148, 2.02257070361929152330437636783, 2.22089217179805665196473659483, 2.29992415055520986592992191184, 2.45860614398229720222162856933, 2.47089798466739842609899493176, 2.47702707615170243842223421271, 2.62271092355865513798338354909, 2.70118135464532415031873530675, 3.11025418437386516967000604337, 3.13635093418087975154235134300, 3.21635398445787068300970969991, 3.37427263367647110572417079281, 3.47899886458601509622706444709, 3.55190946029908857803985925750
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.