Dirichlet series
| L(s) = 1 | − 8·2-s + 36·4-s + 5-s − 120·8-s − 11·9-s − 8·10-s − 6·11-s − 5·13-s + 330·16-s − 9·17-s + 88·18-s − 8·19-s + 36·20-s + 48·22-s − 23-s − 24·25-s + 40·26-s − 27-s − 23·29-s − 6·31-s − 792·32-s + 72·34-s − 396·36-s + 37-s + 64·38-s − 120·40-s − 3·41-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 18·4-s + 0.447·5-s − 42.4·8-s − 3.66·9-s − 2.52·10-s − 1.80·11-s − 1.38·13-s + 82.5·16-s − 2.18·17-s + 20.7·18-s − 1.83·19-s + 8.04·20-s + 10.2·22-s − 0.208·23-s − 4.79·25-s + 7.84·26-s − 0.192·27-s − 4.27·29-s − 1.07·31-s − 140.·32-s + 12.3·34-s − 66·36-s + 0.164·37-s + 10.3·38-s − 18.9·40-s − 0.468·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 19^{8} \cdot 53^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 19^{8} \cdot 53^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 19^{8} \cdot 53^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.47396\times 10^{9}\) |
| Root analytic conductor: | \(4.01022\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{8} \cdot 19^{8} \cdot 53^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | \( ( 1 + T )^{8} \) |
| 19 | \( ( 1 + T )^{8} \) | |
| 53 | \( ( 1 + T )^{8} \) | |
| good | 3 | \( 1 + 11 T^{2} + T^{3} + 68 T^{4} - 2 p T^{5} + 296 T^{6} - 47 T^{7} + 974 T^{8} - 47 p T^{9} + 296 p^{2} T^{10} - 2 p^{4} T^{11} + 68 p^{4} T^{12} + p^{5} T^{13} + 11 p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( 1 - T + p^{2} T^{2} - 18 T^{3} + 298 T^{4} - 36 p T^{5} + 474 p T^{6} - 1327 T^{7} + 13799 T^{8} - 1327 p T^{9} + 474 p^{3} T^{10} - 36 p^{4} T^{11} + 298 p^{4} T^{12} - 18 p^{5} T^{13} + p^{8} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 + 32 T^{2} - 6 T^{3} + 543 T^{4} - 18 p T^{5} + 6189 T^{6} - 1412 T^{7} + 50634 T^{8} - 1412 p T^{9} + 6189 p^{2} T^{10} - 18 p^{4} T^{11} + 543 p^{4} T^{12} - 6 p^{5} T^{13} + 32 p^{6} T^{14} + p^{8} T^{16} \) | |
| 11 | \( 1 + 6 T + 68 T^{2} + 333 T^{3} + 2215 T^{4} + 811 p T^{5} + 4005 p T^{6} + 147507 T^{7} + 586001 T^{8} + 147507 p T^{9} + 4005 p^{3} T^{10} + 811 p^{4} T^{11} + 2215 p^{4} T^{12} + 333 p^{5} T^{13} + 68 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 5 T + 72 T^{2} + 236 T^{3} + 173 p T^{4} + 5788 T^{5} + 47279 T^{6} + 105221 T^{7} + 727088 T^{8} + 105221 p T^{9} + 47279 p^{2} T^{10} + 5788 p^{3} T^{11} + 173 p^{5} T^{12} + 236 p^{5} T^{13} + 72 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 9 T + 93 T^{2} + 599 T^{3} + 3777 T^{4} + 19140 T^{5} + 93237 T^{6} + 405962 T^{7} + 1741443 T^{8} + 405962 p T^{9} + 93237 p^{2} T^{10} + 19140 p^{3} T^{11} + 3777 p^{4} T^{12} + 599 p^{5} T^{13} + 93 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + T + 87 T^{2} + 7 T^{3} + 3981 T^{4} - 2922 T^{5} + 126533 T^{6} - 158176 T^{7} + 3164771 T^{8} - 158176 p T^{9} + 126533 p^{2} T^{10} - 2922 p^{3} T^{11} + 3981 p^{4} T^{12} + 7 p^{5} T^{13} + 87 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 23 T + 392 T^{2} + 4616 T^{3} + 46251 T^{4} + 379914 T^{5} + 2786115 T^{6} + 17648781 T^{7} + 3502988 p T^{8} + 17648781 p T^{9} + 2786115 p^{2} T^{10} + 379914 p^{3} T^{11} + 46251 p^{4} T^{12} + 4616 p^{5} T^{13} + 392 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 6 T + 133 T^{2} + 980 T^{3} + 10256 T^{4} + 70663 T^{5} + 544908 T^{6} + 3168232 T^{7} + 20241471 T^{8} + 3168232 p T^{9} + 544908 p^{2} T^{10} + 70663 p^{3} T^{11} + 10256 p^{4} T^{12} + 980 p^{5} T^{13} + 133 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - T + 177 T^{2} - 271 T^{3} + 14211 T^{4} - 32411 T^{5} + 725352 T^{6} - 2096109 T^{7} + 28953110 T^{8} - 2096109 p T^{9} + 725352 p^{2} T^{10} - 32411 p^{3} T^{11} + 14211 p^{4} T^{12} - 271 p^{5} T^{13} + 177 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 3 T + 175 T^{2} + 693 T^{3} + 14823 T^{4} + 61913 T^{5} + 859270 T^{6} + 3327917 T^{7} + 39028252 T^{8} + 3327917 p T^{9} + 859270 p^{2} T^{10} + 61913 p^{3} T^{11} + 14823 p^{4} T^{12} + 693 p^{5} T^{13} + 175 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 9 T + 315 T^{2} - 2304 T^{3} + 44083 T^{4} - 266855 T^{5} + 3617729 T^{6} - 18118427 T^{7} + 191341153 T^{8} - 18118427 p T^{9} + 3617729 p^{2} T^{10} - 266855 p^{3} T^{11} + 44083 p^{4} T^{12} - 2304 p^{5} T^{13} + 315 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 13 T + 370 T^{2} + 3582 T^{3} + 56587 T^{4} + 432282 T^{5} + 4936577 T^{6} + 30833751 T^{7} + 281106240 T^{8} + 30833751 p T^{9} + 4936577 p^{2} T^{10} + 432282 p^{3} T^{11} + 56587 p^{4} T^{12} + 3582 p^{5} T^{13} + 370 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 9 T + 334 T^{2} + 3178 T^{3} + 56220 T^{4} + 494742 T^{5} + 5971286 T^{6} + 45550131 T^{7} + 425873085 T^{8} + 45550131 p T^{9} + 5971286 p^{2} T^{10} + 494742 p^{3} T^{11} + 56220 p^{4} T^{12} + 3178 p^{5} T^{13} + 334 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 375 T^{2} - 44 T^{3} + 65349 T^{4} - 12955 T^{5} + 7008674 T^{6} - 1548173 T^{7} + 511276146 T^{8} - 1548173 p T^{9} + 7008674 p^{2} T^{10} - 12955 p^{3} T^{11} + 65349 p^{4} T^{12} - 44 p^{5} T^{13} + 375 p^{6} T^{14} + p^{8} T^{16} \) | |
| 67 | \( 1 - 9 T + 339 T^{2} - 2101 T^{3} + 52608 T^{4} - 250486 T^{5} + 5484860 T^{6} - 22192766 T^{7} + 6361008 p T^{8} - 22192766 p T^{9} + 5484860 p^{2} T^{10} - 250486 p^{3} T^{11} + 52608 p^{4} T^{12} - 2101 p^{5} T^{13} + 339 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 31 T + 776 T^{2} + 13186 T^{3} + 199149 T^{4} + 2436836 T^{5} + 27437843 T^{6} + 264865157 T^{7} + 2391709318 T^{8} + 264865157 p T^{9} + 27437843 p^{2} T^{10} + 2436836 p^{3} T^{11} + 199149 p^{4} T^{12} + 13186 p^{5} T^{13} + 776 p^{6} T^{14} + 31 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 5 T + 283 T^{2} - 1203 T^{3} + 39412 T^{4} - 149982 T^{5} + 3867292 T^{6} - 14458774 T^{7} + 308078660 T^{8} - 14458774 p T^{9} + 3867292 p^{2} T^{10} - 149982 p^{3} T^{11} + 39412 p^{4} T^{12} - 1203 p^{5} T^{13} + 283 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 19 T + 549 T^{2} - 7098 T^{3} + 123509 T^{4} - 1266877 T^{5} + 16906653 T^{6} - 144932011 T^{7} + 1589981323 T^{8} - 144932011 p T^{9} + 16906653 p^{2} T^{10} - 1266877 p^{3} T^{11} + 123509 p^{4} T^{12} - 7098 p^{5} T^{13} + 549 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 11 T + 377 T^{2} - 2977 T^{3} + 65229 T^{4} - 384097 T^{5} + 7266342 T^{6} - 34010169 T^{7} + 641688412 T^{8} - 34010169 p T^{9} + 7266342 p^{2} T^{10} - 384097 p^{3} T^{11} + 65229 p^{4} T^{12} - 2977 p^{5} T^{13} + 377 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 41 T + 1059 T^{2} + 19278 T^{3} + 281678 T^{4} + 3397430 T^{5} + 36304056 T^{6} + 354574915 T^{7} + 3381454585 T^{8} + 354574915 p T^{9} + 36304056 p^{2} T^{10} + 3397430 p^{3} T^{11} + 281678 p^{4} T^{12} + 19278 p^{5} T^{13} + 1059 p^{6} T^{14} + 41 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 24 T + 617 T^{2} + 9319 T^{3} + 134608 T^{4} + 1473678 T^{5} + 15650064 T^{6} + 145759027 T^{7} + 1453266943 T^{8} + 145759027 p T^{9} + 15650064 p^{2} T^{10} + 1473678 p^{3} T^{11} + 134608 p^{4} T^{12} + 9319 p^{5} T^{13} + 617 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.24679396582582098625629582816, −3.95402139038303092425826228599, −3.91180094414137372921408093526, −3.64659554048916784300342198307, −3.62405600745704544496742197387, −3.60536077947575259534856408920, −3.57549140405353249429545678779, −3.54820202626243961244588527840, −3.19995719038876122555403409329, −2.76066010464587019615485804265, −2.69870447221768196776104722487, −2.66034596372910487082817222423, −2.62391285118042606134450222628, −2.62275461508078563061659764868, −2.54099080837588885831542887470, −2.42883609165526475185658011732, −2.22238091597994385582363833321, −2.00394082514735710397590276044, −1.87726969988646430836299939738, −1.71186186409129015134099055312, −1.64633589641046309684262953577, −1.57885657593518461367757584885, −1.51646573229821480092274211748, −1.20286233833873250109226866533, −1.01902680210026560204573482019, 0, 0, 0, 0, 0, 0, 0, 0, 1.01902680210026560204573482019, 1.20286233833873250109226866533, 1.51646573229821480092274211748, 1.57885657593518461367757584885, 1.64633589641046309684262953577, 1.71186186409129015134099055312, 1.87726969988646430836299939738, 2.00394082514735710397590276044, 2.22238091597994385582363833321, 2.42883609165526475185658011732, 2.54099080837588885831542887470, 2.62275461508078563061659764868, 2.62391285118042606134450222628, 2.66034596372910487082817222423, 2.69870447221768196776104722487, 2.76066010464587019615485804265, 3.19995719038876122555403409329, 3.54820202626243961244588527840, 3.57549140405353249429545678779, 3.60536077947575259534856408920, 3.62405600745704544496742197387, 3.64659554048916784300342198307, 3.91180094414137372921408093526, 3.95402139038303092425826228599, 4.24679396582582098625629582816
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.