Dirichlet series
| L(s) = 1 | − 8·2-s − 5·3-s + 36·4-s + 2·5-s + 40·6-s − 11·7-s − 120·8-s + 6·9-s − 16·10-s − 180·12-s + 5·13-s + 88·14-s − 10·15-s + 330·16-s + 8·17-s − 48·18-s − 15·19-s + 72·20-s + 55·21-s − 12·23-s + 600·24-s − 6·25-s − 40·26-s + 11·27-s − 396·28-s − 22·29-s + 80·30-s + ⋯ |
| L(s) = 1 | − 5.65·2-s − 2.88·3-s + 18·4-s + 0.894·5-s + 16.3·6-s − 4.15·7-s − 42.4·8-s + 2·9-s − 5.05·10-s − 51.9·12-s + 1.38·13-s + 23.5·14-s − 2.58·15-s + 82.5·16-s + 1.94·17-s − 11.3·18-s − 3.44·19-s + 16.0·20-s + 12.0·21-s − 2.50·23-s + 122.·24-s − 6/5·25-s − 7.84·26-s + 2.11·27-s − 74.8·28-s − 4.08·29-s + 14.6·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 11^{16} \cdot 17^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.35622\times 10^{12}\) |
| Root analytic conductor: | \(5.73153\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{8} \cdot 11^{16} \cdot 17^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | \( ( 1 + T )^{8} \) |
| 11 | \( 1 \) | |
| 17 | \( ( 1 - T )^{8} \) | |
| good | 3 | \( 1 + 5 T + 19 T^{2} + 2 p^{3} T^{3} + 46 p T^{4} + 295 T^{5} + 196 p T^{6} + 1072 T^{7} + 1927 T^{8} + 1072 p T^{9} + 196 p^{3} T^{10} + 295 p^{3} T^{11} + 46 p^{5} T^{12} + 2 p^{8} T^{13} + 19 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 - 2 T + 2 p T^{2} - 19 T^{3} + 89 T^{4} - 27 p T^{5} + 122 p T^{6} - 168 p T^{7} + 596 p T^{8} - 168 p^{2} T^{9} + 122 p^{3} T^{10} - 27 p^{4} T^{11} + 89 p^{4} T^{12} - 19 p^{5} T^{13} + 2 p^{7} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 + 11 T + 88 T^{2} + 489 T^{3} + 2307 T^{4} + 8959 T^{5} + 31258 T^{6} + 95176 T^{7} + 267213 T^{8} + 95176 p T^{9} + 31258 p^{2} T^{10} + 8959 p^{3} T^{11} + 2307 p^{4} T^{12} + 489 p^{5} T^{13} + 88 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 5 T + 44 T^{2} - 211 T^{3} + 1193 T^{4} - 405 p T^{5} + 23758 T^{6} - 88308 T^{7} + 352867 T^{8} - 88308 p T^{9} + 23758 p^{2} T^{10} - 405 p^{4} T^{11} + 1193 p^{4} T^{12} - 211 p^{5} T^{13} + 44 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 15 T + 158 T^{2} + 1155 T^{3} + 7133 T^{4} + 35855 T^{5} + 166306 T^{6} + 695155 T^{7} + 8420 p^{2} T^{8} + 695155 p T^{9} + 166306 p^{2} T^{10} + 35855 p^{3} T^{11} + 7133 p^{4} T^{12} + 1155 p^{5} T^{13} + 158 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 12 T + 6 p T^{2} + 1080 T^{3} + 7485 T^{4} + 44568 T^{5} + 241334 T^{6} + 1218220 T^{7} + 5950339 T^{8} + 1218220 p T^{9} + 241334 p^{2} T^{10} + 44568 p^{3} T^{11} + 7485 p^{4} T^{12} + 1080 p^{5} T^{13} + 6 p^{7} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 22 T + 351 T^{2} + 4224 T^{3} + 41855 T^{4} + 354112 T^{5} + 2599065 T^{6} + 16802290 T^{7} + 96051296 T^{8} + 16802290 p T^{9} + 2599065 p^{2} T^{10} + 354112 p^{3} T^{11} + 41855 p^{4} T^{12} + 4224 p^{5} T^{13} + 351 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 7 T + 52 T^{2} + 177 T^{3} + 51 p T^{4} + 5365 T^{5} + 65604 T^{6} + 385579 T^{7} + 3570244 T^{8} + 385579 p T^{9} + 65604 p^{2} T^{10} + 5365 p^{3} T^{11} + 51 p^{5} T^{12} + 177 p^{5} T^{13} + 52 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 15 T + 263 T^{2} - 2512 T^{3} + 26275 T^{4} - 189422 T^{5} + 1518629 T^{6} - 9121095 T^{7} + 63152528 T^{8} - 9121095 p T^{9} + 1518629 p^{2} T^{10} - 189422 p^{3} T^{11} + 26275 p^{4} T^{12} - 2512 p^{5} T^{13} + 263 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 17 T + 295 T^{2} + 3384 T^{3} + 37671 T^{4} + 336482 T^{5} + 2874933 T^{6} + 506821 p T^{7} + 3498488 p T^{8} + 506821 p^{2} T^{9} + 2874933 p^{2} T^{10} + 336482 p^{3} T^{11} + 37671 p^{4} T^{12} + 3384 p^{5} T^{13} + 295 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 8 T + 131 T^{2} + 718 T^{3} + 9369 T^{4} + 33202 T^{5} + 406437 T^{6} + 895284 T^{7} + 16463628 T^{8} + 895284 p T^{9} + 406437 p^{2} T^{10} + 33202 p^{3} T^{11} + 9369 p^{4} T^{12} + 718 p^{5} T^{13} + 131 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 18 T + 364 T^{2} - 4600 T^{3} + 56799 T^{4} - 556052 T^{5} + 5123896 T^{6} - 40409870 T^{7} + 296786744 T^{8} - 40409870 p T^{9} + 5123896 p^{2} T^{10} - 556052 p^{3} T^{11} + 56799 p^{4} T^{12} - 4600 p^{5} T^{13} + 364 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 20 T + 381 T^{2} - 4393 T^{3} + 51102 T^{4} - 452988 T^{5} + 4286564 T^{6} - 32883294 T^{7} + 266264651 T^{8} - 32883294 p T^{9} + 4286564 p^{2} T^{10} - 452988 p^{3} T^{11} + 51102 p^{4} T^{12} - 4393 p^{5} T^{13} + 381 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 6 T + 264 T^{2} - 1807 T^{3} + 33327 T^{4} - 221201 T^{5} + 2882200 T^{6} - 16342060 T^{7} + 192283936 T^{8} - 16342060 p T^{9} + 2882200 p^{2} T^{10} - 221201 p^{3} T^{11} + 33327 p^{4} T^{12} - 1807 p^{5} T^{13} + 264 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 5 T + 208 T^{2} - 257 T^{3} + 19879 T^{4} + 24625 T^{5} + 1591936 T^{6} + 2143349 T^{7} + 113552752 T^{8} + 2143349 p T^{9} + 1591936 p^{2} T^{10} + 24625 p^{3} T^{11} + 19879 p^{4} T^{12} - 257 p^{5} T^{13} + 208 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 13 T + 359 T^{2} - 3692 T^{3} + 62409 T^{4} - 540222 T^{5} + 6959749 T^{6} - 51323787 T^{7} + 547272212 T^{8} - 51323787 p T^{9} + 6959749 p^{2} T^{10} - 540222 p^{3} T^{11} + 62409 p^{4} T^{12} - 3692 p^{5} T^{13} + 359 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 18 T + 486 T^{2} + 5889 T^{3} + 95617 T^{4} + 914023 T^{5} + 11523810 T^{6} + 1306000 p T^{7} + 970098076 T^{8} + 1306000 p^{2} T^{9} + 11523810 p^{2} T^{10} + 914023 p^{3} T^{11} + 95617 p^{4} T^{12} + 5889 p^{5} T^{13} + 486 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 4 T + 230 T^{2} - 13 T^{3} + 26279 T^{4} + 75903 T^{5} + 2617258 T^{6} + 7180346 T^{7} + 225515360 T^{8} + 7180346 p T^{9} + 2617258 p^{2} T^{10} + 75903 p^{3} T^{11} + 26279 p^{4} T^{12} - 13 p^{5} T^{13} + 230 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 21 T + 671 T^{2} + 10082 T^{3} + 183578 T^{4} + 2151327 T^{5} + 28383902 T^{6} + 268204686 T^{7} + 2777369227 T^{8} + 268204686 p T^{9} + 28383902 p^{2} T^{10} + 2151327 p^{3} T^{11} + 183578 p^{4} T^{12} + 10082 p^{5} T^{13} + 671 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 38 T + 1063 T^{2} + 20480 T^{3} + 338135 T^{4} + 4603692 T^{5} + 56623089 T^{6} + 603921290 T^{7} + 5883979184 T^{8} + 603921290 p T^{9} + 56623089 p^{2} T^{10} + 4603692 p^{3} T^{11} + 338135 p^{4} T^{12} + 20480 p^{5} T^{13} + 1063 p^{6} T^{14} + 38 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 12 T + 559 T^{2} - 5356 T^{3} + 143615 T^{4} - 1140670 T^{5} + 22608545 T^{6} - 150741818 T^{7} + 2410768784 T^{8} - 150741818 p T^{9} + 22608545 p^{2} T^{10} - 1140670 p^{3} T^{11} + 143615 p^{4} T^{12} - 5356 p^{5} T^{13} + 559 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 66 T + 2530 T^{2} + 68829 T^{3} + 1460779 T^{4} + 25300941 T^{5} + 367328690 T^{6} + 4541657652 T^{7} + 48205434880 T^{8} + 4541657652 p T^{9} + 367328690 p^{2} T^{10} + 25300941 p^{3} T^{11} + 1460779 p^{4} T^{12} + 68829 p^{5} T^{13} + 2530 p^{6} T^{14} + 66 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.87117025385688919010026610847, −3.84610994215994652892474988382, −3.50495844318051750154535876884, −3.42062156263570141085792447435, −3.31470810965684019878502928175, −3.18998076278410116876599173113, −3.07879313708800789698459759675, −3.06778593775921059621856594859, −3.00897638814344091020793941638, −2.66585198596456507450568637641, −2.65289849901003207625721653217, −2.42664316061121679852510511271, −2.41470015484913153350356227364, −2.15758881924436918743097113631, −2.15351062527410852122616857801, −2.09050911489310351515647780798, −1.93239181178481550440505287038, −1.88464113631243963328809214702, −1.55925447370982425948571245004, −1.37521804884078900957569274410, −1.32349067328000091556239328597, −1.16873490881932105307016057436, −1.08739922886565067750641066703, −0.993426487700008808261883167701, −0.72098007057494037086739967075, 0, 0, 0, 0, 0, 0, 0, 0, 0.72098007057494037086739967075, 0.993426487700008808261883167701, 1.08739922886565067750641066703, 1.16873490881932105307016057436, 1.32349067328000091556239328597, 1.37521804884078900957569274410, 1.55925447370982425948571245004, 1.88464113631243963328809214702, 1.93239181178481550440505287038, 2.09050911489310351515647780798, 2.15351062527410852122616857801, 2.15758881924436918743097113631, 2.41470015484913153350356227364, 2.42664316061121679852510511271, 2.65289849901003207625721653217, 2.66585198596456507450568637641, 3.00897638814344091020793941638, 3.06778593775921059621856594859, 3.07879313708800789698459759675, 3.18998076278410116876599173113, 3.31470810965684019878502928175, 3.42062156263570141085792447435, 3.50495844318051750154535876884, 3.84610994215994652892474988382, 3.87117025385688919010026610847
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.