Dirichlet series
| L(s) = 1 | − 8·2-s + 3·3-s + 36·4-s + 8·5-s − 24·6-s + 11·7-s − 120·8-s − 6·9-s − 64·10-s − 8·11-s + 108·12-s + 6·13-s − 88·14-s + 24·15-s + 330·16-s + 9·17-s + 48·18-s + 9·19-s + 288·20-s + 33·21-s + 64·22-s − 2·23-s − 360·24-s + 36·25-s − 48·26-s − 22·27-s + 396·28-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 1.73·3-s + 18·4-s + 3.57·5-s − 9.79·6-s + 4.15·7-s − 42.4·8-s − 2·9-s − 20.2·10-s − 2.41·11-s + 31.1·12-s + 1.66·13-s − 23.5·14-s + 6.19·15-s + 82.5·16-s + 2.18·17-s + 11.3·18-s + 2.06·19-s + 64.3·20-s + 7.20·21-s + 13.6·22-s − 0.417·23-s − 73.4·24-s + 36/5·25-s − 9.41·26-s − 4.23·27-s + 74.8·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 11^{8} \cdot 43^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 11^{8} \cdot 43^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 5^{8} \cdot 11^{8} \cdot 43^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.14100\times 10^{12}\) |
| Root analytic conductor: | \(6.14566\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 5^{8} \cdot 11^{8} \cdot 43^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(24.37909323\) |
| \(L(\frac12)\) | \(\approx\) | \(24.37909323\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 - T )^{8} \) | |
| 11 | \( ( 1 + T )^{8} \) | |
| 43 | \( ( 1 + T )^{8} \) | |
| good | 3 | \( 1 - p T + 5 p T^{2} - 41 T^{3} + 118 T^{4} - 89 p T^{5} + 596 T^{6} - 1136 T^{7} + 2099 T^{8} - 1136 p T^{9} + 596 p^{2} T^{10} - 89 p^{4} T^{11} + 118 p^{4} T^{12} - 41 p^{5} T^{13} + 5 p^{7} T^{14} - p^{8} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 11 T + 83 T^{2} - 465 T^{3} + 2140 T^{4} - 8471 T^{5} + 29454 T^{6} - 91492 T^{7} + 255259 T^{8} - 91492 p T^{9} + 29454 p^{2} T^{10} - 8471 p^{3} T^{11} + 2140 p^{4} T^{12} - 465 p^{5} T^{13} + 83 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 6 T + 56 T^{2} - 198 T^{3} + 812 T^{4} - 562 T^{5} - 392 p T^{6} + 52030 T^{7} - 217882 T^{8} + 52030 p T^{9} - 392 p^{3} T^{10} - 562 p^{3} T^{11} + 812 p^{4} T^{12} - 198 p^{5} T^{13} + 56 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 9 T + 87 T^{2} - 601 T^{3} + 3904 T^{4} - 21757 T^{5} + 112362 T^{6} - 522986 T^{7} + 2280891 T^{8} - 522986 p T^{9} + 112362 p^{2} T^{10} - 21757 p^{3} T^{11} + 3904 p^{4} T^{12} - 601 p^{5} T^{13} + 87 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 9 T + 117 T^{2} - 767 T^{3} + 5924 T^{4} - 32115 T^{5} + 186316 T^{6} - 864248 T^{7} + 4124941 T^{8} - 864248 p T^{9} + 186316 p^{2} T^{10} - 32115 p^{3} T^{11} + 5924 p^{4} T^{12} - 767 p^{5} T^{13} + 117 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 2 T + 96 T^{2} + 126 T^{3} + 156 p T^{4} - 818 T^{5} + 67120 T^{6} - 218798 T^{7} + 1067670 T^{8} - 218798 p T^{9} + 67120 p^{2} T^{10} - 818 p^{3} T^{11} + 156 p^{5} T^{12} + 126 p^{5} T^{13} + 96 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 10 T + 189 T^{2} + 1466 T^{3} + 16362 T^{4} + 104346 T^{5} + 861411 T^{6} + 4575426 T^{7} + 30229322 T^{8} + 4575426 p T^{9} + 861411 p^{2} T^{10} + 104346 p^{3} T^{11} + 16362 p^{4} T^{12} + 1466 p^{5} T^{13} + 189 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 6 T + 169 T^{2} - 842 T^{3} + 13550 T^{4} - 58090 T^{5} + 695239 T^{6} - 2593702 T^{7} + 25290434 T^{8} - 2593702 p T^{9} + 695239 p^{2} T^{10} - 58090 p^{3} T^{11} + 13550 p^{4} T^{12} - 842 p^{5} T^{13} + 169 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 16 T + 321 T^{2} - 3518 T^{3} + 41066 T^{4} - 344554 T^{5} + 2954111 T^{6} - 19874912 T^{7} + 134955402 T^{8} - 19874912 p T^{9} + 2954111 p^{2} T^{10} - 344554 p^{3} T^{11} + 41066 p^{4} T^{12} - 3518 p^{5} T^{13} + 321 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 4 T + 184 T^{2} + 944 T^{3} + 17684 T^{4} + 98904 T^{5} + 1151320 T^{6} + 6165204 T^{7} + 54805686 T^{8} + 6165204 p T^{9} + 1151320 p^{2} T^{10} + 98904 p^{3} T^{11} + 17684 p^{4} T^{12} + 944 p^{5} T^{13} + 184 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 11 T + 290 T^{2} + 2678 T^{3} + 40016 T^{4} + 308551 T^{5} + 3390883 T^{6} + 21891232 T^{7} + 192450008 T^{8} + 21891232 p T^{9} + 3390883 p^{2} T^{10} + 308551 p^{3} T^{11} + 40016 p^{4} T^{12} + 2678 p^{5} T^{13} + 290 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 15 T + 359 T^{2} + 4191 T^{3} + 58688 T^{4} + 550765 T^{5} + 5725974 T^{6} + 44240966 T^{7} + 368327611 T^{8} + 44240966 p T^{9} + 5725974 p^{2} T^{10} + 550765 p^{3} T^{11} + 58688 p^{4} T^{12} + 4191 p^{5} T^{13} + 359 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 7 T + 310 T^{2} + 1870 T^{3} + 48406 T^{4} + 256023 T^{5} + 4863943 T^{6} + 22258856 T^{7} + 340193048 T^{8} + 22258856 p T^{9} + 4863943 p^{2} T^{10} + 256023 p^{3} T^{11} + 48406 p^{4} T^{12} + 1870 p^{5} T^{13} + 310 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 2 T + 173 T^{2} - 258 T^{3} + 13146 T^{4} - 48466 T^{5} + 722187 T^{6} - 6226362 T^{7} + 40704730 T^{8} - 6226362 p T^{9} + 722187 p^{2} T^{10} - 48466 p^{3} T^{11} + 13146 p^{4} T^{12} - 258 p^{5} T^{13} + 173 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 4 T + 296 T^{2} - 1092 T^{3} + 39900 T^{4} - 144516 T^{5} + 3458392 T^{6} - 12643748 T^{7} + 242751078 T^{8} - 12643748 p T^{9} + 3458392 p^{2} T^{10} - 144516 p^{3} T^{11} + 39900 p^{4} T^{12} - 1092 p^{5} T^{13} + 296 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 3 T + 251 T^{2} + 817 T^{3} + 34524 T^{4} + 118515 T^{5} + 3534502 T^{6} + 12628708 T^{7} + 283024187 T^{8} + 12628708 p T^{9} + 3534502 p^{2} T^{10} + 118515 p^{3} T^{11} + 34524 p^{4} T^{12} + 817 p^{5} T^{13} + 251 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 32 T + 704 T^{2} - 10628 T^{3} + 137164 T^{4} - 1496452 T^{5} + 15474608 T^{6} - 144995000 T^{7} + 1303399494 T^{8} - 144995000 p T^{9} + 15474608 p^{2} T^{10} - 1496452 p^{3} T^{11} + 137164 p^{4} T^{12} - 10628 p^{5} T^{13} + 704 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 45 T + 16 p T^{2} - 24352 T^{3} + 368254 T^{4} - 4481047 T^{5} + 47083561 T^{6} - 442336508 T^{7} + 3997142488 T^{8} - 442336508 p T^{9} + 47083561 p^{2} T^{10} - 4481047 p^{3} T^{11} + 368254 p^{4} T^{12} - 24352 p^{5} T^{13} + 16 p^{7} T^{14} - 45 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 39 T + 902 T^{2} - 14594 T^{3} + 198374 T^{4} - 2399223 T^{5} + 27273811 T^{6} - 284037900 T^{7} + 2724078224 T^{8} - 284037900 p T^{9} + 27273811 p^{2} T^{10} - 2399223 p^{3} T^{11} + 198374 p^{4} T^{12} - 14594 p^{5} T^{13} + 902 p^{6} T^{14} - 39 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 22 T + 489 T^{2} + 7894 T^{3} + 127750 T^{4} + 1616582 T^{5} + 19871335 T^{6} + 209703870 T^{7} + 2155136002 T^{8} + 209703870 p T^{9} + 19871335 p^{2} T^{10} + 1616582 p^{3} T^{11} + 127750 p^{4} T^{12} + 7894 p^{5} T^{13} + 489 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 30 T + 972 T^{2} - 18646 T^{3} + 355644 T^{4} - 5057418 T^{5} + 70504772 T^{6} - 786673474 T^{7} + 8582427558 T^{8} - 786673474 p T^{9} + 70504772 p^{2} T^{10} - 5057418 p^{3} T^{11} + 355644 p^{4} T^{12} - 18646 p^{5} T^{13} + 972 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.33678463760419593860687093283, −3.04266655583077253477251905977, −2.86829719642425185448687729073, −2.85157808800020088141213145531, −2.77251647160644092422696359503, −2.59270507190925843132144340455, −2.57332048930387464723874315319, −2.52692939246473177063689615272, −2.50925676878965534525717311398, −1.93992957939802800159259125210, −1.93211249159987812812064471908, −1.91227089197938808324933482417, −1.87578837574000246223892711698, −1.85046617272342965768742068651, −1.83818736282055131117462868579, −1.70725619319663960912269870228, −1.59755424076671576687763726925, −1.34055934603123096723275137641, −1.14262104522903966403114556769, −0.938969928687791974355523244310, −0.850864147653843924695134022360, −0.62096796357383007756299827365, −0.59890707293714280763495544046, −0.52662309714559379852450488702, −0.51596464537423928475349764277, 0.51596464537423928475349764277, 0.52662309714559379852450488702, 0.59890707293714280763495544046, 0.62096796357383007756299827365, 0.850864147653843924695134022360, 0.938969928687791974355523244310, 1.14262104522903966403114556769, 1.34055934603123096723275137641, 1.59755424076671576687763726925, 1.70725619319663960912269870228, 1.83818736282055131117462868579, 1.85046617272342965768742068651, 1.87578837574000246223892711698, 1.91227089197938808324933482417, 1.93211249159987812812064471908, 1.93992957939802800159259125210, 2.50925676878965534525717311398, 2.52692939246473177063689615272, 2.57332048930387464723874315319, 2.59270507190925843132144340455, 2.77251647160644092422696359503, 2.85157808800020088141213145531, 2.86829719642425185448687729073, 3.04266655583077253477251905977, 3.33678463760419593860687093283
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.