Dirichlet series
| L(s) = 1 | − 2-s + 3·4-s + 5-s + 3·7-s − 2·8-s − 10-s + 3·11-s + 9·13-s − 3·14-s + 6·16-s + 8·17-s − 14·19-s + 3·20-s − 3·22-s + 10·23-s + 15·25-s − 9·26-s + 9·28-s − 15·29-s + 15·31-s − 3·32-s − 8·34-s + 3·35-s − 24·37-s + 14·38-s − 2·40-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 3/2·4-s + 0.447·5-s + 1.13·7-s − 0.707·8-s − 0.316·10-s + 0.904·11-s + 2.49·13-s − 0.801·14-s + 3/2·16-s + 1.94·17-s − 3.21·19-s + 0.670·20-s − 0.639·22-s + 2.08·23-s + 3·25-s − 1.76·26-s + 1.70·28-s − 2.78·29-s + 2.69·31-s − 0.530·32-s − 1.37·34-s + 0.507·35-s − 3.94·37-s + 2.27·38-s − 0.316·40-s − 0.937·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{24} \cdot 17^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(32562.5\) |
| Root analytic conductor: | \(1.91445\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{24} \cdot 17^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(11.58446408\) |
| \(L(\frac12)\) | \(\approx\) | \(11.58446408\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 17 | \( ( 1 - T )^{8} \) | |
| good | 2 | \( ( 1 - T + T^{4} - p^{3} T^{7} + p^{4} T^{8} )( 1 + p T - 3 T^{3} - 5 T^{4} - 3 p T^{5} + p^{4} T^{7} + p^{4} T^{8} ) \) |
| 5 | \( 1 - T - 14 T^{2} + 9 T^{3} + 22 p T^{4} - 34 T^{5} - 711 T^{6} + 64 T^{7} + 3889 T^{8} + 64 p T^{9} - 711 p^{2} T^{10} - 34 p^{3} T^{11} + 22 p^{5} T^{12} + 9 p^{5} T^{13} - 14 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 - 3 T - 2 p T^{2} + 39 T^{3} + 136 T^{4} - 234 T^{5} - 1277 T^{6} + 72 p T^{7} + 1543 p T^{8} + 72 p^{2} T^{9} - 1277 p^{2} T^{10} - 234 p^{3} T^{11} + 136 p^{4} T^{12} + 39 p^{5} T^{13} - 2 p^{7} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 3 T - 26 T^{2} + 39 T^{3} + 490 T^{4} - 18 p T^{5} - 6887 T^{6} + 1374 T^{7} + 73429 T^{8} + 1374 p T^{9} - 6887 p^{2} T^{10} - 18 p^{4} T^{11} + 490 p^{4} T^{12} + 39 p^{5} T^{13} - 26 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 9 T + p T^{2} + 42 T^{3} + 274 T^{4} - 1161 T^{5} - 4022 T^{6} - 5154 T^{7} + 135469 T^{8} - 5154 p T^{9} - 4022 p^{2} T^{10} - 1161 p^{3} T^{11} + 274 p^{4} T^{12} + 42 p^{5} T^{13} + p^{7} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( ( 1 + 7 T + 34 T^{2} + 181 T^{3} + 1021 T^{4} + 181 p T^{5} + 34 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 23 | \( 1 - 10 T + T^{2} + 186 T^{3} + 584 T^{4} - 4996 T^{5} - 15117 T^{6} + 70216 T^{7} + 152287 T^{8} + 70216 p T^{9} - 15117 p^{2} T^{10} - 4996 p^{3} T^{11} + 584 p^{4} T^{12} + 186 p^{5} T^{13} + p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 15 T + 43 T^{2} - 60 T^{3} + 3298 T^{4} + 27315 T^{5} + 25342 T^{6} + 353040 T^{7} + 5203561 T^{8} + 353040 p T^{9} + 25342 p^{2} T^{10} + 27315 p^{3} T^{11} + 3298 p^{4} T^{12} - 60 p^{5} T^{13} + 43 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 15 T + 25 T^{2} + 90 T^{3} + 6250 T^{4} - 39645 T^{5} - 2000 p T^{6} - 510240 T^{7} + 11875279 T^{8} - 510240 p T^{9} - 2000 p^{3} T^{10} - 39645 p^{3} T^{11} + 6250 p^{4} T^{12} + 90 p^{5} T^{13} + 25 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( ( 1 + 12 T + 113 T^{2} + 756 T^{3} + 4401 T^{4} + 756 p T^{5} + 113 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 41 | \( 1 + 6 T - 83 T^{2} - 726 T^{3} + 3292 T^{4} + 36612 T^{5} - 37973 T^{6} - 696720 T^{7} - 33089 T^{8} - 696720 p T^{9} - 37973 p^{2} T^{10} + 36612 p^{3} T^{11} + 3292 p^{4} T^{12} - 726 p^{5} T^{13} - 83 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 18 T + 148 T^{2} - 396 T^{3} - 3794 T^{4} + 50058 T^{5} - 229592 T^{6} + 76446 T^{7} + 4695967 T^{8} + 76446 p T^{9} - 229592 p^{2} T^{10} + 50058 p^{3} T^{11} - 3794 p^{4} T^{12} - 396 p^{5} T^{13} + 148 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 12 T - 80 T^{2} + 642 T^{3} + 13729 T^{4} - 56007 T^{5} - 865811 T^{6} + 457017 T^{7} + 55050073 T^{8} + 457017 p T^{9} - 865811 p^{2} T^{10} - 56007 p^{3} T^{11} + 13729 p^{4} T^{12} + 642 p^{5} T^{13} - 80 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( ( 1 + 12 T + 227 T^{2} + 1854 T^{3} + 18363 T^{4} + 1854 p T^{5} + 227 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 59 | \( 1 - 122 T^{2} + 576 T^{3} + 6937 T^{4} - 52128 T^{5} - 37226 T^{6} + 1857312 T^{7} - 5667452 T^{8} + 1857312 p T^{9} - 37226 p^{2} T^{10} - 52128 p^{3} T^{11} + 6937 p^{4} T^{12} + 576 p^{5} T^{13} - 122 p^{6} T^{14} + p^{8} T^{16} \) | |
| 61 | \( 1 + 5 T - 60 T^{2} - 419 T^{3} - 3616 T^{4} - 25698 T^{5} - 60191 T^{6} + 2024672 T^{7} + 35986221 T^{8} + 2024672 p T^{9} - 60191 p^{2} T^{10} - 25698 p^{3} T^{11} - 3616 p^{4} T^{12} - 419 p^{5} T^{13} - 60 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 6 T - 155 T^{2} + 1134 T^{3} + 12136 T^{4} - 84060 T^{5} - 787337 T^{6} + 2272488 T^{7} + 57092431 T^{8} + 2272488 p T^{9} - 787337 p^{2} T^{10} - 84060 p^{3} T^{11} + 12136 p^{4} T^{12} + 1134 p^{5} T^{13} - 155 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( ( 1 - 25 T + 447 T^{2} - 5406 T^{3} + 52285 T^{4} - 5406 p T^{5} + 447 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 73 | \( ( 1 - 4 T + 197 T^{2} - 624 T^{3} + 18077 T^{4} - 624 p T^{5} + 197 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 79 | \( 1 - 8 T - 130 T^{2} + 698 T^{3} + 11315 T^{4} - 29989 T^{5} - 384141 T^{6} + 1367031 T^{7} - 10694699 T^{8} + 1367031 p T^{9} - 384141 p^{2} T^{10} - 29989 p^{3} T^{11} + 11315 p^{4} T^{12} + 698 p^{5} T^{13} - 130 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 7 T - 275 T^{2} - 942 T^{3} + 53966 T^{4} + 96385 T^{5} - 6779508 T^{6} - 2857684 T^{7} + 653637691 T^{8} - 2857684 p T^{9} - 6779508 p^{2} T^{10} + 96385 p^{3} T^{11} + 53966 p^{4} T^{12} - 942 p^{5} T^{13} - 275 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 - 14 T + 222 T^{2} - 1497 T^{3} + 19543 T^{4} - 1497 p T^{5} + 222 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 16 T - 54 T^{2} + 1216 T^{3} + 9869 T^{4} - 78480 T^{5} - 838382 T^{6} + 10731488 T^{7} - 94479444 T^{8} + 10731488 p T^{9} - 838382 p^{2} T^{10} - 78480 p^{3} T^{11} + 9869 p^{4} T^{12} + 1216 p^{5} T^{13} - 54 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.93171647480171449157683942365, −4.90841085169579196012743064453, −4.79516413746234643723416101105, −4.47413148250163163136288544454, −4.25984999635466343890365571917, −4.01587055210668990051508292810, −3.94337318674106459856057783989, −3.90166903066062545372918324129, −3.50354575943590664500187029090, −3.47633934041869629922681002225, −3.43628509904122098467565825816, −3.37162292705752369026138711949, −3.19415967908606623446494848732, −2.92446823674689734247702427699, −2.48662375311980226648159929474, −2.34334423402296457673357168216, −2.24133685810284811443527369266, −2.16750311388098496041637439001, −2.15655325921897698987464211530, −1.76336391590306534050067657172, −1.24810661714817408080059299928, −1.23449264519445715526081079561, −1.16122918885508434248688389246, −1.02622750530232039876315096333, −0.64875590801532541949849456975, 0.64875590801532541949849456975, 1.02622750530232039876315096333, 1.16122918885508434248688389246, 1.23449264519445715526081079561, 1.24810661714817408080059299928, 1.76336391590306534050067657172, 2.15655325921897698987464211530, 2.16750311388098496041637439001, 2.24133685810284811443527369266, 2.34334423402296457673357168216, 2.48662375311980226648159929474, 2.92446823674689734247702427699, 3.19415967908606623446494848732, 3.37162292705752369026138711949, 3.43628509904122098467565825816, 3.47633934041869629922681002225, 3.50354575943590664500187029090, 3.90166903066062545372918324129, 3.94337318674106459856057783989, 4.01587055210668990051508292810, 4.25984999635466343890365571917, 4.47413148250163163136288544454, 4.79516413746234643723416101105, 4.90841085169579196012743064453, 4.93171647480171449157683942365
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.