Dirichlet series
| L(s) = 1 | + 4·2-s + 438·4-s + 840·5-s − 140·7-s + 232·8-s + 3.36e3·10-s − 1.78e3·11-s − 560·14-s + 9.82e4·16-s + 1.41e5·17-s − 2.57e5·19-s + 3.67e5·20-s − 7.13e3·22-s + 3.48e5·23-s − 7.07e5·25-s − 6.13e4·28-s − 4.98e6·29-s − 2.37e6·31-s − 3.49e5·32-s + 5.65e5·34-s − 1.17e5·35-s + 4.92e5·37-s − 1.03e6·38-s + 1.94e5·40-s + 4.44e6·43-s − 7.81e5·44-s + 1.39e6·46-s + ⋯ |
| L(s) = 1 | + 1/4·2-s + 1.71·4-s + 1.34·5-s − 0.0583·7-s + 0.0566·8-s + 0.335·10-s − 0.121·11-s − 0.0145·14-s + 1.49·16-s + 1.69·17-s − 1.97·19-s + 2.29·20-s − 0.0304·22-s + 1.24·23-s − 1.81·25-s − 0.0997·28-s − 7.04·29-s − 2.57·31-s − 0.333·32-s + 0.423·34-s − 0.0783·35-s + 0.262·37-s − 0.494·38-s + 0.0761·40-s + 1.30·43-s − 0.208·44-s + 0.311·46-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+4)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{16} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.88239\times 10^{11}\) |
| Root analytic conductor: | \(5.06604\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{16} \cdot 7^{8} ,\ ( \ : [4]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(0.1282189676\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1282189676\) |
| \(L(5)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 20 p T + 5584 p^{3} T^{2} + 34180 p^{6} T^{3} - 146722 p^{10} T^{4} + 34180 p^{14} T^{5} + 5584 p^{19} T^{6} + 20 p^{25} T^{7} + p^{32} T^{8} \) | |
| good | 2 | \( 1 - p^{2} T - 211 p T^{2} + 401 p^{3} T^{3} + 18669 p^{2} T^{4} - 26973 p^{5} T^{5} + 438931 p^{5} T^{6} + 422859 p^{8} T^{7} - 25012319 p^{8} T^{8} + 422859 p^{16} T^{9} + 438931 p^{21} T^{10} - 26973 p^{29} T^{11} + 18669 p^{34} T^{12} + 401 p^{43} T^{13} - 211 p^{49} T^{14} - p^{58} T^{15} + p^{64} T^{16} \) |
| 5 | \( 1 - 168 p T + 1412926 T^{2} - 197857968 p T^{3} + 963491077201 T^{4} - 91893714315984 p T^{5} + 15265912316977486 p^{2} T^{6} - 1133093210050933656 p^{3} T^{7} + \)\(20\!\cdots\!56\)\( p^{4} T^{8} - 1133093210050933656 p^{11} T^{9} + 15265912316977486 p^{18} T^{10} - 91893714315984 p^{25} T^{11} + 963491077201 p^{32} T^{12} - 197857968 p^{41} T^{13} + 1412926 p^{48} T^{14} - 168 p^{57} T^{15} + p^{64} T^{16} \) | |
| 11 | \( 1 + 1784 T - 333848516 T^{2} - 1417907526152 T^{3} - 3397673438524281 T^{4} + 13244519048971696380 p T^{5} - \)\(72\!\cdots\!96\)\( T^{6} + \)\(15\!\cdots\!72\)\( T^{7} + \)\(57\!\cdots\!72\)\( T^{8} + \)\(15\!\cdots\!72\)\( p^{8} T^{9} - \)\(72\!\cdots\!96\)\( p^{16} T^{10} + 13244519048971696380 p^{25} T^{11} - 3397673438524281 p^{32} T^{12} - 1417907526152 p^{40} T^{13} - 333848516 p^{48} T^{14} + 1784 p^{56} T^{15} + p^{64} T^{16} \) | |
| 13 | \( 1 - 3599704928 T^{2} + 6293879827435246396 T^{4} - \)\(73\!\cdots\!48\)\( T^{6} + \)\(65\!\cdots\!34\)\( T^{8} - \)\(73\!\cdots\!48\)\( p^{16} T^{10} + 6293879827435246396 p^{32} T^{12} - 3599704928 p^{48} T^{14} + p^{64} T^{16} \) | |
| 17 | \( 1 - 141456 T + 24752467462 T^{2} - 2557882950722400 T^{3} + \)\(31\!\cdots\!53\)\( T^{4} - \)\(34\!\cdots\!68\)\( T^{5} + \)\(33\!\cdots\!54\)\( T^{6} - \)\(19\!\cdots\!28\)\( p T^{7} + \)\(86\!\cdots\!16\)\( p^{2} T^{8} - \)\(19\!\cdots\!28\)\( p^{9} T^{9} + \)\(33\!\cdots\!54\)\( p^{16} T^{10} - \)\(34\!\cdots\!68\)\( p^{24} T^{11} + \)\(31\!\cdots\!53\)\( p^{32} T^{12} - 2557882950722400 p^{40} T^{13} + 24752467462 p^{48} T^{14} - 141456 p^{56} T^{15} + p^{64} T^{16} \) | |
| 19 | \( 1 + 257544 T + 70009367908 T^{2} + 12336288216616224 T^{3} + \)\(20\!\cdots\!39\)\( T^{4} + \)\(30\!\cdots\!20\)\( T^{5} + \)\(43\!\cdots\!52\)\( T^{6} + \)\(61\!\cdots\!88\)\( T^{7} + \)\(82\!\cdots\!52\)\( T^{8} + \)\(61\!\cdots\!88\)\( p^{8} T^{9} + \)\(43\!\cdots\!52\)\( p^{16} T^{10} + \)\(30\!\cdots\!20\)\( p^{24} T^{11} + \)\(20\!\cdots\!39\)\( p^{32} T^{12} + 12336288216616224 p^{40} T^{13} + 70009367908 p^{48} T^{14} + 257544 p^{56} T^{15} + p^{64} T^{16} \) | |
| 23 | \( 1 - 348940 T - 186978426644 T^{2} + 32676900622844200 T^{3} + \)\(35\!\cdots\!19\)\( T^{4} - \)\(26\!\cdots\!40\)\( T^{5} - \)\(40\!\cdots\!80\)\( T^{6} + \)\(10\!\cdots\!80\)\( T^{7} + \)\(34\!\cdots\!72\)\( T^{8} + \)\(10\!\cdots\!80\)\( p^{8} T^{9} - \)\(40\!\cdots\!80\)\( p^{16} T^{10} - \)\(26\!\cdots\!40\)\( p^{24} T^{11} + \)\(35\!\cdots\!19\)\( p^{32} T^{12} + 32676900622844200 p^{40} T^{13} - 186978426644 p^{48} T^{14} - 348940 p^{56} T^{15} + p^{64} T^{16} \) | |
| 29 | \( ( 1 + 2491588 T + 3930639559056 T^{2} + 4191570786881848748 T^{3} + \)\(34\!\cdots\!34\)\( T^{4} + 4191570786881848748 p^{8} T^{5} + 3930639559056 p^{16} T^{6} + 2491588 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 31 | \( 1 + 2376696 T + 4495851115156 T^{2} + 6210203237206004064 T^{3} + \)\(70\!\cdots\!63\)\( T^{4} + \)\(75\!\cdots\!64\)\( T^{5} + \)\(78\!\cdots\!20\)\( T^{6} + \)\(78\!\cdots\!92\)\( T^{7} + \)\(76\!\cdots\!08\)\( T^{8} + \)\(78\!\cdots\!92\)\( p^{8} T^{9} + \)\(78\!\cdots\!20\)\( p^{16} T^{10} + \)\(75\!\cdots\!64\)\( p^{24} T^{11} + \)\(70\!\cdots\!63\)\( p^{32} T^{12} + 6210203237206004064 p^{40} T^{13} + 4495851115156 p^{48} T^{14} + 2376696 p^{56} T^{15} + p^{64} T^{16} \) | |
| 37 | \( 1 - 492740 T - 6839200540314 T^{2} + 5004808393655458680 T^{3} + \)\(16\!\cdots\!69\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{5} - \)\(40\!\cdots\!30\)\( T^{6} + \)\(11\!\cdots\!20\)\( T^{7} + \)\(20\!\cdots\!12\)\( T^{8} + \)\(11\!\cdots\!20\)\( p^{8} T^{9} - \)\(40\!\cdots\!30\)\( p^{16} T^{10} - \)\(13\!\cdots\!40\)\( p^{24} T^{11} + \)\(16\!\cdots\!69\)\( p^{32} T^{12} + 5004808393655458680 p^{40} T^{13} - 6839200540314 p^{48} T^{14} - 492740 p^{56} T^{15} + p^{64} T^{16} \) | |
| 41 | \( 1 - 37601617521056 T^{2} + \)\(67\!\cdots\!60\)\( T^{4} - \)\(78\!\cdots\!52\)\( T^{6} + \)\(69\!\cdots\!34\)\( T^{8} - \)\(78\!\cdots\!52\)\( p^{16} T^{10} + \)\(67\!\cdots\!60\)\( p^{32} T^{12} - 37601617521056 p^{48} T^{14} + p^{64} T^{16} \) | |
| 43 | \( ( 1 - 2224216 T + 36058463751268 T^{2} - 59838646062515811848 T^{3} + \)\(58\!\cdots\!34\)\( T^{4} - 59838646062515811848 p^{8} T^{5} + 36058463751268 p^{16} T^{6} - 2224216 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 47 | \( 1 + 2704128 T + 43354016357404 T^{2} + \)\(11\!\cdots\!28\)\( T^{3} + \)\(86\!\cdots\!35\)\( T^{4} - \)\(26\!\cdots\!60\)\( T^{5} - \)\(76\!\cdots\!24\)\( T^{6} - \)\(20\!\cdots\!24\)\( T^{7} - \)\(69\!\cdots\!32\)\( T^{8} - \)\(20\!\cdots\!24\)\( p^{8} T^{9} - \)\(76\!\cdots\!24\)\( p^{16} T^{10} - \)\(26\!\cdots\!60\)\( p^{24} T^{11} + \)\(86\!\cdots\!35\)\( p^{32} T^{12} + \)\(11\!\cdots\!28\)\( p^{40} T^{13} + 43354016357404 p^{48} T^{14} + 2704128 p^{56} T^{15} + p^{64} T^{16} \) | |
| 53 | \( 1 + 2281460 T - 194027845315466 T^{2} - \)\(48\!\cdots\!80\)\( T^{3} + \)\(21\!\cdots\!21\)\( T^{4} + \)\(44\!\cdots\!40\)\( T^{5} - \)\(17\!\cdots\!38\)\( T^{6} - \)\(13\!\cdots\!20\)\( T^{7} + \)\(11\!\cdots\!08\)\( T^{8} - \)\(13\!\cdots\!20\)\( p^{8} T^{9} - \)\(17\!\cdots\!38\)\( p^{16} T^{10} + \)\(44\!\cdots\!40\)\( p^{24} T^{11} + \)\(21\!\cdots\!21\)\( p^{32} T^{12} - \)\(48\!\cdots\!80\)\( p^{40} T^{13} - 194027845315466 p^{48} T^{14} + 2281460 p^{56} T^{15} + p^{64} T^{16} \) | |
| 59 | \( 1 + 25291140 T + 442662123701212 T^{2} + \)\(58\!\cdots\!80\)\( T^{3} + \)\(47\!\cdots\!71\)\( T^{4} + \)\(33\!\cdots\!24\)\( p T^{5} - \)\(50\!\cdots\!36\)\( T^{6} - \)\(24\!\cdots\!88\)\( p T^{7} - \)\(20\!\cdots\!44\)\( T^{8} - \)\(24\!\cdots\!88\)\( p^{9} T^{9} - \)\(50\!\cdots\!36\)\( p^{16} T^{10} + \)\(33\!\cdots\!24\)\( p^{25} T^{11} + \)\(47\!\cdots\!71\)\( p^{32} T^{12} + \)\(58\!\cdots\!80\)\( p^{40} T^{13} + 442662123701212 p^{48} T^{14} + 25291140 p^{56} T^{15} + p^{64} T^{16} \) | |
| 61 | \( 1 - 59368764 T + 2052606459885622 T^{2} - \)\(52\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!05\)\( T^{4} - \)\(18\!\cdots\!72\)\( T^{5} + \)\(29\!\cdots\!94\)\( T^{6} - \)\(44\!\cdots\!08\)\( T^{7} + \)\(62\!\cdots\!84\)\( T^{8} - \)\(44\!\cdots\!08\)\( p^{8} T^{9} + \)\(29\!\cdots\!94\)\( p^{16} T^{10} - \)\(18\!\cdots\!72\)\( p^{24} T^{11} + \)\(10\!\cdots\!05\)\( p^{32} T^{12} - \)\(52\!\cdots\!60\)\( p^{40} T^{13} + 2052606459885622 p^{48} T^{14} - 59368764 p^{56} T^{15} + p^{64} T^{16} \) | |
| 67 | \( 1 + 107108 T - 839204332190988 T^{2} - \)\(18\!\cdots\!44\)\( T^{3} + \)\(37\!\cdots\!55\)\( T^{4} + \)\(11\!\cdots\!96\)\( T^{5} + \)\(84\!\cdots\!16\)\( T^{6} - \)\(32\!\cdots\!36\)\( T^{7} - \)\(68\!\cdots\!16\)\( T^{8} - \)\(32\!\cdots\!36\)\( p^{8} T^{9} + \)\(84\!\cdots\!16\)\( p^{16} T^{10} + \)\(11\!\cdots\!96\)\( p^{24} T^{11} + \)\(37\!\cdots\!55\)\( p^{32} T^{12} - \)\(18\!\cdots\!44\)\( p^{40} T^{13} - 839204332190988 p^{48} T^{14} + 107108 p^{56} T^{15} + p^{64} T^{16} \) | |
| 71 | \( ( 1 - 41404880 T + 3162773608457988 T^{2} - \)\(83\!\cdots\!72\)\( T^{3} + \)\(32\!\cdots\!98\)\( T^{4} - \)\(83\!\cdots\!72\)\( p^{8} T^{5} + 3162773608457988 p^{16} T^{6} - 41404880 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 73 | \( 1 - 116758404 T + 8148090587057374 T^{2} - \)\(42\!\cdots\!08\)\( T^{3} + \)\(17\!\cdots\!45\)\( T^{4} - \)\(65\!\cdots\!68\)\( T^{5} + \)\(21\!\cdots\!86\)\( T^{6} - \)\(66\!\cdots\!72\)\( T^{7} + \)\(19\!\cdots\!68\)\( T^{8} - \)\(66\!\cdots\!72\)\( p^{8} T^{9} + \)\(21\!\cdots\!86\)\( p^{16} T^{10} - \)\(65\!\cdots\!68\)\( p^{24} T^{11} + \)\(17\!\cdots\!45\)\( p^{32} T^{12} - \)\(42\!\cdots\!08\)\( p^{40} T^{13} + 8148090587057374 p^{48} T^{14} - 116758404 p^{56} T^{15} + p^{64} T^{16} \) | |
| 79 | \( 1 + 50628092 T - 1644037081623204 T^{2} - \)\(20\!\cdots\!48\)\( T^{3} + \)\(47\!\cdots\!03\)\( T^{4} - \)\(41\!\cdots\!00\)\( T^{5} - \)\(54\!\cdots\!80\)\( T^{6} - \)\(48\!\cdots\!40\)\( T^{7} + \)\(31\!\cdots\!48\)\( T^{8} - \)\(48\!\cdots\!40\)\( p^{8} T^{9} - \)\(54\!\cdots\!80\)\( p^{16} T^{10} - \)\(41\!\cdots\!00\)\( p^{24} T^{11} + \)\(47\!\cdots\!03\)\( p^{32} T^{12} - \)\(20\!\cdots\!48\)\( p^{40} T^{13} - 1644037081623204 p^{48} T^{14} + 50628092 p^{56} T^{15} + p^{64} T^{16} \) | |
| 83 | \( 1 - 11214933300710504 T^{2} + \)\(52\!\cdots\!64\)\( T^{4} - \)\(14\!\cdots\!32\)\( T^{6} + \)\(31\!\cdots\!18\)\( T^{8} - \)\(14\!\cdots\!32\)\( p^{16} T^{10} + \)\(52\!\cdots\!64\)\( p^{32} T^{12} - 11214933300710504 p^{48} T^{14} + p^{64} T^{16} \) | |
| 89 | \( 1 - 2322516 T + 13958973359482918 T^{2} - \)\(32\!\cdots\!56\)\( T^{3} + \)\(11\!\cdots\!33\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{5} + \)\(67\!\cdots\!14\)\( T^{6} - \)\(46\!\cdots\!92\)\( T^{7} + \)\(30\!\cdots\!76\)\( T^{8} - \)\(46\!\cdots\!92\)\( p^{8} T^{9} + \)\(67\!\cdots\!14\)\( p^{16} T^{10} - \)\(13\!\cdots\!20\)\( p^{24} T^{11} + \)\(11\!\cdots\!33\)\( p^{32} T^{12} - \)\(32\!\cdots\!56\)\( p^{40} T^{13} + 13958973359482918 p^{48} T^{14} - 2322516 p^{56} T^{15} + p^{64} T^{16} \) | |
| 97 | \( 1 - 32582515803200672 T^{2} + \)\(61\!\cdots\!96\)\( T^{4} - \)\(75\!\cdots\!32\)\( T^{6} + \)\(68\!\cdots\!18\)\( T^{8} - \)\(75\!\cdots\!32\)\( p^{16} T^{10} + \)\(61\!\cdots\!96\)\( p^{32} T^{12} - 32582515803200672 p^{48} T^{14} + p^{64} T^{16} \) | |
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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
−5.38467798891946454687488226129, −5.23000858763943641789999803852, −5.08205453478554337479577333292, −5.03724816952544612634179635793, −4.43251195695610577430580508109, −4.03530483150811415465039762785, −3.93242785792261516100382710925, −3.79630535024482131841836168811, −3.76827321699673441068681132334, −3.60836635391029080462906907513, −3.50393589531968725379404224560, −3.04530938590518421220760597035, −2.77603997474551272695031971203, −2.48673111015471526994791617498, −2.39827209571471353275837997701, −2.11377506043082775027715752950, −2.06486768423839919518734336016, −1.80279449423021457519409792760, −1.72225293813018319667410282821, −1.56667349706255735697786446725, −1.25490077260312514622945537802, −0.954476813159799845151682239632, −0.65335967338269096022691856134, −0.096379140141234127967248281346, −0.07118489217515746682494629973, 0.07118489217515746682494629973, 0.096379140141234127967248281346, 0.65335967338269096022691856134, 0.954476813159799845151682239632, 1.25490077260312514622945537802, 1.56667349706255735697786446725, 1.72225293813018319667410282821, 1.80279449423021457519409792760, 2.06486768423839919518734336016, 2.11377506043082775027715752950, 2.39827209571471353275837997701, 2.48673111015471526994791617498, 2.77603997474551272695031971203, 3.04530938590518421220760597035, 3.50393589531968725379404224560, 3.60836635391029080462906907513, 3.76827321699673441068681132334, 3.79630535024482131841836168811, 3.93242785792261516100382710925, 4.03530483150811415465039762785, 4.43251195695610577430580508109, 5.03724816952544612634179635793, 5.08205453478554337479577333292, 5.23000858763943641789999803852, 5.38467798891946454687488226129
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.