Dirichlet series
| L(s) = 1 | − 8·2-s − 4·3-s + 36·4-s + 6·5-s + 32·6-s − 8·7-s − 120·8-s − 48·10-s + 4·11-s − 144·12-s − 6·13-s + 64·14-s − 24·15-s + 330·16-s + 2·17-s + 216·20-s + 32·21-s − 32·22-s + 20·23-s + 480·24-s − 6·25-s + 48·26-s + 18·27-s − 288·28-s − 16·29-s + 192·30-s − 22·31-s + ⋯ |
| L(s) = 1 | − 5.65·2-s − 2.30·3-s + 18·4-s + 2.68·5-s + 13.0·6-s − 3.02·7-s − 42.4·8-s − 15.1·10-s + 1.20·11-s − 41.5·12-s − 1.66·13-s + 17.1·14-s − 6.19·15-s + 82.5·16-s + 0.485·17-s + 48.2·20-s + 6.98·21-s − 6.82·22-s + 4.17·23-s + 97.9·24-s − 6/5·25-s + 9.41·26-s + 3.46·27-s − 54.4·28-s − 2.97·29-s + 35.0·30-s − 3.95·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 7^{8} \cdot 19^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.03556\times 10^{12}\) |
| Root analytic conductor: | \(6.35266\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{8} \cdot 7^{8} \cdot 19^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | \( ( 1 + T )^{8} \) |
| 7 | \( ( 1 + T )^{8} \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + 4 T + 16 T^{2} + 46 T^{3} + 41 p T^{4} + 284 T^{5} + 614 T^{6} + 1208 T^{7} + 2173 T^{8} + 1208 p T^{9} + 614 p^{2} T^{10} + 284 p^{3} T^{11} + 41 p^{5} T^{12} + 46 p^{5} T^{13} + 16 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 - 6 T + 42 T^{2} - 168 T^{3} + 142 p T^{4} - 2182 T^{5} + 6872 T^{6} - 16934 T^{7} + 42391 T^{8} - 16934 p T^{9} + 6872 p^{2} T^{10} - 2182 p^{3} T^{11} + 142 p^{5} T^{12} - 168 p^{5} T^{13} + 42 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 4 T + 60 T^{2} - 240 T^{3} + 1750 T^{4} - 6852 T^{5} + 32568 T^{6} - 117180 T^{7} + 423015 T^{8} - 117180 p T^{9} + 32568 p^{2} T^{10} - 6852 p^{3} T^{11} + 1750 p^{4} T^{12} - 240 p^{5} T^{13} + 60 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 6 T + 66 T^{2} + 28 p T^{3} + 2123 T^{4} + 9886 T^{5} + 44274 T^{6} + 170392 T^{7} + 666613 T^{8} + 170392 p T^{9} + 44274 p^{2} T^{10} + 9886 p^{3} T^{11} + 2123 p^{4} T^{12} + 28 p^{6} T^{13} + 66 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 2 T + 94 T^{2} - 152 T^{3} + 4323 T^{4} - 346 p T^{5} + 126346 T^{6} - 145596 T^{7} + 2552893 T^{8} - 145596 p T^{9} + 126346 p^{2} T^{10} - 346 p^{4} T^{11} + 4323 p^{4} T^{12} - 152 p^{5} T^{13} + 94 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 20 T + 264 T^{2} - 120 p T^{3} + 23662 T^{4} - 173980 T^{5} + 1122872 T^{6} - 6399700 T^{7} + 32483735 T^{8} - 6399700 p T^{9} + 1122872 p^{2} T^{10} - 173980 p^{3} T^{11} + 23662 p^{4} T^{12} - 120 p^{6} T^{13} + 264 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 16 T + 244 T^{2} + 2496 T^{3} + 814 p T^{4} + 181728 T^{5} + 1303976 T^{6} + 8034448 T^{7} + 46404967 T^{8} + 8034448 p T^{9} + 1303976 p^{2} T^{10} + 181728 p^{3} T^{11} + 814 p^{5} T^{12} + 2496 p^{5} T^{13} + 244 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 22 T + 406 T^{2} + 5048 T^{3} + 55126 T^{4} + 483446 T^{5} + 3804224 T^{6} + 25295434 T^{7} + 152252047 T^{8} + 25295434 p T^{9} + 3804224 p^{2} T^{10} + 483446 p^{3} T^{11} + 55126 p^{4} T^{12} + 5048 p^{5} T^{13} + 406 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 12 T + 184 T^{2} - 1312 T^{3} + 13198 T^{4} - 77532 T^{5} + 19288 p T^{6} - 3913756 T^{7} + 31155703 T^{8} - 3913756 p T^{9} + 19288 p^{3} T^{10} - 77532 p^{3} T^{11} + 13198 p^{4} T^{12} - 1312 p^{5} T^{13} + 184 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 36 T + 780 T^{2} + 12110 T^{3} + 150395 T^{4} + 1554328 T^{5} + 13818578 T^{6} + 106988980 T^{7} + 729875985 T^{8} + 106988980 p T^{9} + 13818578 p^{2} T^{10} + 1554328 p^{3} T^{11} + 150395 p^{4} T^{12} + 12110 p^{5} T^{13} + 780 p^{6} T^{14} + 36 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 154 T^{2} - 80 T^{3} + 9967 T^{4} - 14640 T^{5} + 367172 T^{6} - 1185040 T^{7} + 12372685 T^{8} - 1185040 p T^{9} + 367172 p^{2} T^{10} - 14640 p^{3} T^{11} + 9967 p^{4} T^{12} - 80 p^{5} T^{13} + 154 p^{6} T^{14} + p^{8} T^{16} \) | |
| 47 | \( 1 - 6 T + 298 T^{2} - 1972 T^{3} + 41206 T^{4} - 274146 T^{5} + 3486920 T^{6} - 21233246 T^{7} + 197841639 T^{8} - 21233246 p T^{9} + 3486920 p^{2} T^{10} - 274146 p^{3} T^{11} + 41206 p^{4} T^{12} - 1972 p^{5} T^{13} + 298 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 16 T + 376 T^{2} + 4424 T^{3} + 61478 T^{4} + 580136 T^{5} + 5996984 T^{6} + 46678232 T^{7} + 386446463 T^{8} + 46678232 p T^{9} + 5996984 p^{2} T^{10} + 580136 p^{3} T^{11} + 61478 p^{4} T^{12} + 4424 p^{5} T^{13} + 376 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 14 T + 254 T^{2} + 2804 T^{3} + 35986 T^{4} + 334642 T^{5} + 3457656 T^{6} + 27278342 T^{7} + 233821827 T^{8} + 27278342 p T^{9} + 3457656 p^{2} T^{10} + 334642 p^{3} T^{11} + 35986 p^{4} T^{12} + 2804 p^{5} T^{13} + 254 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 10 T + 318 T^{2} - 2620 T^{3} + 48238 T^{4} - 349430 T^{5} + 4834736 T^{6} - 30830310 T^{7} + 347107735 T^{8} - 30830310 p T^{9} + 4834736 p^{2} T^{10} - 349430 p^{3} T^{11} + 48238 p^{4} T^{12} - 2620 p^{5} T^{13} + 318 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 20 T + 406 T^{2} - 5380 T^{3} + 69247 T^{4} - 738900 T^{5} + 7634548 T^{6} - 68823040 T^{7} + 600554185 T^{8} - 68823040 p T^{9} + 7634548 p^{2} T^{10} - 738900 p^{3} T^{11} + 69247 p^{4} T^{12} - 5380 p^{5} T^{13} + 406 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 328 T^{2} - 320 T^{3} + 52308 T^{4} - 101120 T^{5} + 5523096 T^{6} - 13497280 T^{7} + 440891110 T^{8} - 13497280 p T^{9} + 5523096 p^{2} T^{10} - 101120 p^{3} T^{11} + 52308 p^{4} T^{12} - 320 p^{5} T^{13} + 328 p^{6} T^{14} + p^{8} T^{16} \) | |
| 73 | \( 1 + 18 T + 582 T^{2} + 8184 T^{3} + 146331 T^{4} + 1658678 T^{5} + 21073450 T^{6} + 194166912 T^{7} + 1913088689 T^{8} + 194166912 p T^{9} + 21073450 p^{2} T^{10} + 1658678 p^{3} T^{11} + 146331 p^{4} T^{12} + 8184 p^{5} T^{13} + 582 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 4 T + 294 T^{2} + 1304 T^{3} + 44091 T^{4} + 226612 T^{5} + 4767436 T^{6} + 26192272 T^{7} + 416122757 T^{8} + 26192272 p T^{9} + 4767436 p^{2} T^{10} + 226612 p^{3} T^{11} + 44091 p^{4} T^{12} + 1304 p^{5} T^{13} + 294 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 36 T + 836 T^{2} - 13604 T^{3} + 186368 T^{4} - 26252 p T^{5} + 23799084 T^{6} - 238118692 T^{7} + 2268501198 T^{8} - 238118692 p T^{9} + 23799084 p^{2} T^{10} - 26252 p^{4} T^{11} + 186368 p^{4} T^{12} - 13604 p^{5} T^{13} + 836 p^{6} T^{14} - 36 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 18 T + 610 T^{2} + 9020 T^{3} + 168235 T^{4} + 2071474 T^{5} + 27710822 T^{6} + 284522220 T^{7} + 3001802745 T^{8} + 284522220 p T^{9} + 27710822 p^{2} T^{10} + 2071474 p^{3} T^{11} + 168235 p^{4} T^{12} + 9020 p^{5} T^{13} + 610 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 26 T + 718 T^{2} + 12472 T^{3} + 216306 T^{4} + 2897546 T^{5} + 38440800 T^{6} + 419299866 T^{7} + 4513191899 T^{8} + 419299866 p T^{9} + 38440800 p^{2} T^{10} + 2897546 p^{3} T^{11} + 216306 p^{4} T^{12} + 12472 p^{5} T^{13} + 718 p^{6} T^{14} + 26 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.69209286039870280576288508766, −3.50644218616088226840988884440, −3.37761842069032244358601122067, −3.20327688353235480636277826408, −3.17846215226253392511414201757, −3.14693366017679181593184571761, −3.14142248137414945968510268977, −3.03100948467862119741942205117, −2.72877327553472561548860397880, −2.53330982820828622500315270941, −2.49657439214453847037448106132, −2.42054364181598371391313598794, −2.21365308701668306501813740112, −2.15602963833778506186201527813, −2.12321489713658621020458110610, −2.04396013153723085000488433514, −1.94809101400451791790309806313, −1.74201463138740928877591924282, −1.46224688366918979002958018604, −1.42690565439226072803949189922, −1.33576941047208968213458936862, −1.12760386961375626487535621492, −1.06628290621900129370446298099, −1.03498072620691723709401347237, −0.890744655897504245259592941337, 0, 0, 0, 0, 0, 0, 0, 0, 0.890744655897504245259592941337, 1.03498072620691723709401347237, 1.06628290621900129370446298099, 1.12760386961375626487535621492, 1.33576941047208968213458936862, 1.42690565439226072803949189922, 1.46224688366918979002958018604, 1.74201463138740928877591924282, 1.94809101400451791790309806313, 2.04396013153723085000488433514, 2.12321489713658621020458110610, 2.15602963833778506186201527813, 2.21365308701668306501813740112, 2.42054364181598371391313598794, 2.49657439214453847037448106132, 2.53330982820828622500315270941, 2.72877327553472561548860397880, 3.03100948467862119741942205117, 3.14142248137414945968510268977, 3.14693366017679181593184571761, 3.17846215226253392511414201757, 3.20327688353235480636277826408, 3.37761842069032244358601122067, 3.50644218616088226840988884440, 3.69209286039870280576288508766
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.