Dirichlet series
| L(s) = 1 | − 8·2-s + 36·4-s − 8·5-s + 8·7-s − 120·8-s − 8·9-s + 64·10-s + 13-s − 64·14-s + 330·16-s − 6·17-s + 64·18-s − 5·19-s − 288·20-s + 10·23-s + 36·25-s − 8·26-s + 288·28-s − 3·29-s − 8·31-s − 792·32-s + 48·34-s − 64·35-s − 288·36-s − 6·37-s + 40·38-s + 960·40-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 18·4-s − 3.57·5-s + 3.02·7-s − 42.4·8-s − 8/3·9-s + 20.2·10-s + 0.277·13-s − 17.1·14-s + 82.5·16-s − 1.45·17-s + 15.0·18-s − 1.14·19-s − 64.3·20-s + 2.08·23-s + 36/5·25-s − 1.56·26-s + 54.4·28-s − 0.557·29-s − 1.43·31-s − 140.·32-s + 8.23·34-s − 10.8·35-s − 48·36-s − 0.986·37-s + 6.48·38-s + 151.·40-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 5^{8} \cdot 7^{8} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.37808\times 10^{14}\) |
| Root analytic conductor: | \(8.22394\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{8} \cdot 5^{8} \cdot 7^{8} \cdot 11^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{8} \) |
| 5 | \( ( 1 + T )^{8} \) | |
| 7 | \( ( 1 - T )^{8} \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + 8 T^{2} + 11 p T^{4} - 10 T^{5} + 110 T^{6} - 80 T^{7} + 341 T^{8} - 80 p T^{9} + 110 p^{2} T^{10} - 10 p^{3} T^{11} + 11 p^{5} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - T + 17 T^{2} + 49 T^{3} + 162 T^{4} + 701 T^{5} + 2811 T^{6} + 10103 T^{7} + 23142 T^{8} + 10103 p T^{9} + 2811 p^{2} T^{10} + 701 p^{3} T^{11} + 162 p^{4} T^{12} + 49 p^{5} T^{13} + 17 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 6 T + 74 T^{2} + 364 T^{3} + 2517 T^{4} + 584 p T^{5} + 55138 T^{6} + 186138 T^{7} + 980057 T^{8} + 186138 p T^{9} + 55138 p^{2} T^{10} + 584 p^{4} T^{11} + 2517 p^{4} T^{12} + 364 p^{5} T^{13} + 74 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 5 T + 46 T^{2} + 180 T^{3} + 82 p T^{4} + 7065 T^{5} + 45063 T^{6} + 159420 T^{7} + 887004 T^{8} + 159420 p T^{9} + 45063 p^{2} T^{10} + 7065 p^{3} T^{11} + 82 p^{5} T^{12} + 180 p^{5} T^{13} + 46 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 10 T + 148 T^{2} - 1130 T^{3} + 9268 T^{4} - 59090 T^{5} + 350700 T^{6} - 1945810 T^{7} + 9369686 T^{8} - 1945810 p T^{9} + 350700 p^{2} T^{10} - 59090 p^{3} T^{11} + 9268 p^{4} T^{12} - 1130 p^{5} T^{13} + 148 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 3 T + 119 T^{2} + 299 T^{3} + 6620 T^{4} + 15955 T^{5} + 253525 T^{6} + 618047 T^{7} + 7942994 T^{8} + 618047 p T^{9} + 253525 p^{2} T^{10} + 15955 p^{3} T^{11} + 6620 p^{4} T^{12} + 299 p^{5} T^{13} + 119 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 8 T + 128 T^{2} + 386 T^{3} + 4792 T^{4} - 1878 T^{5} + 153964 T^{6} - 83560 T^{7} + 6250342 T^{8} - 83560 p T^{9} + 153964 p^{2} T^{10} - 1878 p^{3} T^{11} + 4792 p^{4} T^{12} + 386 p^{5} T^{13} + 128 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 6 T + 166 T^{2} + 1112 T^{3} + 13056 T^{4} + 89704 T^{5} + 690310 T^{6} + 4474010 T^{7} + 28407366 T^{8} + 4474010 p T^{9} + 690310 p^{2} T^{10} + 89704 p^{3} T^{11} + 13056 p^{4} T^{12} + 1112 p^{5} T^{13} + 166 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 11 T + 124 T^{2} + 878 T^{3} + 6428 T^{4} + 22765 T^{5} + 100545 T^{6} - 298878 T^{7} - 1904064 T^{8} - 298878 p T^{9} + 100545 p^{2} T^{10} + 22765 p^{3} T^{11} + 6428 p^{4} T^{12} + 878 p^{5} T^{13} + 124 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 5 T + 156 T^{2} - 1112 T^{3} + 14252 T^{4} - 2209 p T^{5} + 958199 T^{6} - 5367826 T^{7} + 47078316 T^{8} - 5367826 p T^{9} + 958199 p^{2} T^{10} - 2209 p^{4} T^{11} + 14252 p^{4} T^{12} - 1112 p^{5} T^{13} + 156 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 15 T + 275 T^{2} + 2525 T^{3} + 28426 T^{4} + 213045 T^{5} + 2042525 T^{6} + 14025095 T^{7} + 114368906 T^{8} + 14025095 p T^{9} + 2042525 p^{2} T^{10} + 213045 p^{3} T^{11} + 28426 p^{4} T^{12} + 2525 p^{5} T^{13} + 275 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 16 T + 192 T^{2} + 1966 T^{3} + 21900 T^{4} + 193178 T^{5} + 1606988 T^{6} + 12594320 T^{7} + 98924158 T^{8} + 12594320 p T^{9} + 1606988 p^{2} T^{10} + 193178 p^{3} T^{11} + 21900 p^{4} T^{12} + 1966 p^{5} T^{13} + 192 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 9 T + 234 T^{2} + 2104 T^{3} + 34306 T^{4} + 269797 T^{5} + 3244791 T^{6} + 22625212 T^{7} + 226891572 T^{8} + 22625212 p T^{9} + 3244791 p^{2} T^{10} + 269797 p^{3} T^{11} + 34306 p^{4} T^{12} + 2104 p^{5} T^{13} + 234 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 32 T + 824 T^{2} + 14626 T^{3} + 222408 T^{4} + 2765794 T^{5} + 496748 p T^{6} + 284999148 T^{7} + 2388069542 T^{8} + 284999148 p T^{9} + 496748 p^{3} T^{10} + 2765794 p^{3} T^{11} + 222408 p^{4} T^{12} + 14626 p^{5} T^{13} + 824 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 33 T + 842 T^{2} - 15042 T^{3} + 229138 T^{4} - 2884273 T^{5} + 32190375 T^{6} - 311931366 T^{7} + 2722654900 T^{8} - 311931366 p T^{9} + 32190375 p^{2} T^{10} - 2884273 p^{3} T^{11} + 229138 p^{4} T^{12} - 15042 p^{5} T^{13} + 842 p^{6} T^{14} - 33 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 11 T + 425 T^{2} - 4759 T^{3} + 86016 T^{4} - 913881 T^{5} + 10870055 T^{6} - 102265729 T^{7} + 929735530 T^{8} - 102265729 p T^{9} + 10870055 p^{2} T^{10} - 913881 p^{3} T^{11} + 86016 p^{4} T^{12} - 4759 p^{5} T^{13} + 425 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 34 T + 752 T^{2} - 11904 T^{3} + 158025 T^{4} - 1775712 T^{5} + 18078418 T^{6} - 167305460 T^{7} + 1476137933 T^{8} - 167305460 p T^{9} + 18078418 p^{2} T^{10} - 1775712 p^{3} T^{11} + 158025 p^{4} T^{12} - 11904 p^{5} T^{13} + 752 p^{6} T^{14} - 34 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 31 T + 713 T^{2} + 10199 T^{3} + 119396 T^{4} + 957285 T^{5} + 6373271 T^{6} + 24224753 T^{7} + 156969274 T^{8} + 24224753 p T^{9} + 6373271 p^{2} T^{10} + 957285 p^{3} T^{11} + 119396 p^{4} T^{12} + 10199 p^{5} T^{13} + 713 p^{6} T^{14} + 31 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 50 T + 1544 T^{2} + 33386 T^{3} + 572973 T^{4} + 8071520 T^{5} + 98073518 T^{6} + 1045485168 T^{7} + 10042941577 T^{8} + 1045485168 p T^{9} + 98073518 p^{2} T^{10} + 8071520 p^{3} T^{11} + 572973 p^{4} T^{12} + 33386 p^{5} T^{13} + 1544 p^{6} T^{14} + 50 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - T + 504 T^{2} - 1162 T^{3} + 117472 T^{4} - 392061 T^{5} + 17106935 T^{6} - 63616750 T^{7} + 1769275296 T^{8} - 63616750 p T^{9} + 17106935 p^{2} T^{10} - 392061 p^{3} T^{11} + 117472 p^{4} T^{12} - 1162 p^{5} T^{13} + 504 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 4 T + 392 T^{2} + 1000 T^{3} + 71413 T^{4} + 145422 T^{5} + 9028218 T^{6} + 19286276 T^{7} + 945132769 T^{8} + 19286276 p T^{9} + 9028218 p^{2} T^{10} + 145422 p^{3} T^{11} + 71413 p^{4} T^{12} + 1000 p^{5} T^{13} + 392 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.48641904288138276072217843134, −3.28068264940698269598165466902, −3.22915052538416672294512313351, −3.04171055652687104135099270987, −3.00882798014600572229618743726, −2.96267211292272941229171106203, −2.89317522561712016039164328629, −2.71307841838782901749547365503, −2.65919820798311876204164338615, −2.43130750594572662766739361489, −2.35397432140423889900980526664, −2.20329386492109827194972727999, −2.18705871193823950219741341454, −2.12710702051247901221840878649, −2.05198506485032231518406313796, −1.73827978017805004058323021901, −1.65833667611238168708439514274, −1.61684801675764245775817978615, −1.38870508801184222864486358109, −1.18866781335755356244009245842, −1.13123680950631186273727471385, −1.10267600141168974818253999764, −1.07761383722983084508320568114, −0.978945424295542839997379250892, −0.874912103265861320843057207188, 0, 0, 0, 0, 0, 0, 0, 0, 0.874912103265861320843057207188, 0.978945424295542839997379250892, 1.07761383722983084508320568114, 1.10267600141168974818253999764, 1.13123680950631186273727471385, 1.18866781335755356244009245842, 1.38870508801184222864486358109, 1.61684801675764245775817978615, 1.65833667611238168708439514274, 1.73827978017805004058323021901, 2.05198506485032231518406313796, 2.12710702051247901221840878649, 2.18705871193823950219741341454, 2.20329386492109827194972727999, 2.35397432140423889900980526664, 2.43130750594572662766739361489, 2.65919820798311876204164338615, 2.71307841838782901749547365503, 2.89317522561712016039164328629, 2.96267211292272941229171106203, 3.00882798014600572229618743726, 3.04171055652687104135099270987, 3.22915052538416672294512313351, 3.28068264940698269598165466902, 3.48641904288138276072217843134
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.