Dirichlet series
| L(s) = 1 | + 1.31e5·3-s + 1.01e10·7-s + 1.56e11·9-s + 5.05e13·13-s + 9.78e14·19-s + 1.34e15·21-s + 2.42e17·25-s + 4.87e16·27-s − 3.09e18·31-s + 2.25e19·37-s + 6.66e18·39-s − 2.26e20·43-s − 6.62e20·49-s + 1.28e20·57-s − 6.50e21·61-s + 1.58e21·63-s + 7.04e21·67-s + 3.62e22·73-s + 3.19e22·75-s − 3.16e23·79-s − 5.11e22·81-s + 5.13e23·91-s − 4.07e23·93-s + 8.16e23·97-s + 1.76e24·103-s + 8.92e24·109-s + 2.97e24·111-s + ⋯ |
| L(s) = 1 | + 0.248·3-s + 0.734·7-s + 0.553·9-s + 2.17·13-s + 0.441·19-s + 0.182·21-s + 4.06·25-s + 0.324·27-s − 3.92·31-s + 3.43·37-s + 0.538·39-s − 5.66·43-s − 3.45·49-s + 0.109·57-s − 2.45·61-s + 0.406·63-s + 0.860·67-s + 1.58·73-s + 1.00·75-s − 5.36·79-s − 0.641·81-s + 1.59·91-s − 0.974·93-s + 1.17·97-s + 1.23·103-s + 3.17·109-s + 0.851·111-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(25-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+12)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 3^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.87074\times 10^{17}\) |
| Root analytic conductor: | \(13.2357\) |
| Motivic weight: | \(24\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 3^{8} ,\ ( \ : [12]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{25}{2})\) | \(\approx\) | \(63.31811503\) |
| \(L(\frac12)\) | \(\approx\) | \(63.31811503\) |
| \(L(13)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 43960 p T - 190519012 p^{6} T^{2} - 6136572680 p^{13} T^{3} + 2567030625142 p^{22} T^{4} - 6136572680 p^{37} T^{5} - 190519012 p^{54} T^{6} - 43960 p^{73} T^{7} + p^{96} T^{8} \) | |
| good | 5 | \( 1 - 242233888485069512 T^{2} + \)\(16\!\cdots\!76\)\( p^{3} T^{4} - \)\(42\!\cdots\!12\)\( p^{6} T^{6} + \)\(89\!\cdots\!26\)\( p^{14} T^{8} - \)\(42\!\cdots\!12\)\( p^{54} T^{10} + \)\(16\!\cdots\!76\)\( p^{99} T^{12} - 242233888485069512 p^{144} T^{14} + p^{192} T^{16} \) |
| 7 | \( ( 1 - 5080397320 T + 52842286586751462436 p T^{2} - \)\(12\!\cdots\!40\)\( p^{3} T^{3} + \)\(56\!\cdots\!02\)\( p^{6} T^{4} - \)\(12\!\cdots\!40\)\( p^{27} T^{5} + 52842286586751462436 p^{49} T^{6} - 5080397320 p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 11 | \( 1 - \)\(43\!\cdots\!80\)\( T^{2} + \)\(87\!\cdots\!44\)\( p^{2} T^{4} - \)\(95\!\cdots\!40\)\( p^{6} T^{6} + \)\(74\!\cdots\!66\)\( p^{10} T^{8} - \)\(95\!\cdots\!40\)\( p^{54} T^{10} + \)\(87\!\cdots\!44\)\( p^{98} T^{12} - \)\(43\!\cdots\!80\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 13 | \( ( 1 - 25284181839560 T + \)\(94\!\cdots\!44\)\( T^{2} - \)\(30\!\cdots\!60\)\( p T^{3} + \)\(17\!\cdots\!54\)\( p^{2} T^{4} - \)\(30\!\cdots\!60\)\( p^{25} T^{5} + \)\(94\!\cdots\!44\)\( p^{48} T^{6} - 25284181839560 p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 17 | \( 1 - \)\(15\!\cdots\!60\)\( T^{2} + \)\(44\!\cdots\!16\)\( p^{2} T^{4} - \)\(85\!\cdots\!80\)\( p^{4} T^{6} + \)\(11\!\cdots\!74\)\( p^{6} T^{8} - \)\(85\!\cdots\!80\)\( p^{52} T^{10} + \)\(44\!\cdots\!16\)\( p^{98} T^{12} - \)\(15\!\cdots\!60\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 19 | \( ( 1 - 489041815670632 T + \)\(87\!\cdots\!12\)\( p T^{2} - \)\(22\!\cdots\!84\)\( p^{2} T^{3} + \)\(16\!\cdots\!70\)\( p^{3} T^{4} - \)\(22\!\cdots\!84\)\( p^{26} T^{5} + \)\(87\!\cdots\!12\)\( p^{49} T^{6} - 489041815670632 p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 23 | \( 1 - \)\(23\!\cdots\!20\)\( p^{2} T^{2} + \)\(41\!\cdots\!24\)\( p^{4} T^{4} - \)\(56\!\cdots\!60\)\( p^{6} T^{6} + \)\(54\!\cdots\!06\)\( p^{8} T^{8} - \)\(56\!\cdots\!60\)\( p^{54} T^{10} + \)\(41\!\cdots\!24\)\( p^{100} T^{12} - \)\(23\!\cdots\!20\)\( p^{146} T^{14} + p^{192} T^{16} \) | |
| 29 | \( 1 - \)\(36\!\cdots\!88\)\( T^{2} + \)\(15\!\cdots\!48\)\( T^{4} + \)\(40\!\cdots\!04\)\( T^{6} + \)\(13\!\cdots\!70\)\( T^{8} + \)\(40\!\cdots\!04\)\( p^{48} T^{10} + \)\(15\!\cdots\!48\)\( p^{96} T^{12} - \)\(36\!\cdots\!88\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 31 | \( ( 1 + 1546059893411354552 T + \)\(25\!\cdots\!48\)\( T^{2} + \)\(21\!\cdots\!64\)\( T^{3} + \)\(21\!\cdots\!70\)\( T^{4} + \)\(21\!\cdots\!64\)\( p^{24} T^{5} + \)\(25\!\cdots\!48\)\( p^{48} T^{6} + 1546059893411354552 p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 37 | \( ( 1 - 11295146611996391240 T + \)\(17\!\cdots\!52\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!38\)\( T^{4} - \)\(12\!\cdots\!60\)\( p^{24} T^{5} + \)\(17\!\cdots\!52\)\( p^{48} T^{6} - 11295146611996391240 p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 41 | \( 1 - \)\(20\!\cdots\!40\)\( T^{2} + \)\(23\!\cdots\!24\)\( T^{4} - \)\(18\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!26\)\( T^{8} - \)\(18\!\cdots\!20\)\( p^{48} T^{10} + \)\(23\!\cdots\!24\)\( p^{96} T^{12} - \)\(20\!\cdots\!40\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 43 | \( ( 1 + \)\(11\!\cdots\!40\)\( T + \)\(89\!\cdots\!36\)\( T^{2} + \)\(47\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!46\)\( T^{4} + \)\(47\!\cdots\!00\)\( p^{24} T^{5} + \)\(89\!\cdots\!36\)\( p^{48} T^{6} + \)\(11\!\cdots\!40\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 47 | \( 1 - \)\(68\!\cdots\!48\)\( T^{2} + \)\(23\!\cdots\!08\)\( T^{4} - \)\(54\!\cdots\!96\)\( T^{6} + \)\(86\!\cdots\!70\)\( T^{8} - \)\(54\!\cdots\!96\)\( p^{48} T^{10} + \)\(23\!\cdots\!08\)\( p^{96} T^{12} - \)\(68\!\cdots\!48\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 53 | \( 1 - \)\(59\!\cdots\!80\)\( T^{2} + \)\(23\!\cdots\!04\)\( T^{4} - \)\(73\!\cdots\!40\)\( T^{6} + \)\(17\!\cdots\!46\)\( T^{8} - \)\(73\!\cdots\!40\)\( p^{48} T^{10} + \)\(23\!\cdots\!04\)\( p^{96} T^{12} - \)\(59\!\cdots\!80\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 59 | \( 1 - \)\(12\!\cdots\!80\)\( T^{2} + \)\(79\!\cdots\!44\)\( T^{4} - \)\(35\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!66\)\( T^{8} - \)\(35\!\cdots\!40\)\( p^{48} T^{10} + \)\(79\!\cdots\!44\)\( p^{96} T^{12} - \)\(12\!\cdots\!80\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 61 | \( ( 1 + \)\(32\!\cdots\!24\)\( T + \)\(16\!\cdots\!20\)\( T^{2} + \)\(41\!\cdots\!96\)\( T^{3} + \)\(13\!\cdots\!14\)\( T^{4} + \)\(41\!\cdots\!96\)\( p^{24} T^{5} + \)\(16\!\cdots\!20\)\( p^{48} T^{6} + \)\(32\!\cdots\!24\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 67 | \( ( 1 - \)\(35\!\cdots\!60\)\( T + \)\(20\!\cdots\!12\)\( T^{2} - \)\(42\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!98\)\( T^{4} - \)\(42\!\cdots\!00\)\( p^{24} T^{5} + \)\(20\!\cdots\!12\)\( p^{48} T^{6} - \)\(35\!\cdots\!60\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 71 | \( 1 - \)\(10\!\cdots\!88\)\( T^{2} + \)\(63\!\cdots\!48\)\( T^{4} - \)\(26\!\cdots\!96\)\( T^{6} + \)\(83\!\cdots\!70\)\( T^{8} - \)\(26\!\cdots\!96\)\( p^{48} T^{10} + \)\(63\!\cdots\!48\)\( p^{96} T^{12} - \)\(10\!\cdots\!88\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 73 | \( ( 1 - \)\(18\!\cdots\!60\)\( T + \)\(72\!\cdots\!24\)\( T^{2} - \)\(40\!\cdots\!80\)\( T^{3} + \)\(36\!\cdots\!06\)\( T^{4} - \)\(40\!\cdots\!80\)\( p^{24} T^{5} + \)\(72\!\cdots\!24\)\( p^{48} T^{6} - \)\(18\!\cdots\!60\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 79 | \( ( 1 + \)\(15\!\cdots\!12\)\( T + \)\(21\!\cdots\!88\)\( T^{2} + \)\(17\!\cdots\!84\)\( T^{3} + \)\(12\!\cdots\!30\)\( T^{4} + \)\(17\!\cdots\!84\)\( p^{24} T^{5} + \)\(21\!\cdots\!88\)\( p^{48} T^{6} + \)\(15\!\cdots\!12\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 83 | \( 1 - \)\(34\!\cdots\!76\)\( T^{2} + \)\(78\!\cdots\!20\)\( T^{4} - \)\(13\!\cdots\!04\)\( T^{6} + \)\(23\!\cdots\!26\)\( p^{2} T^{8} - \)\(13\!\cdots\!04\)\( p^{48} T^{10} + \)\(78\!\cdots\!20\)\( p^{96} T^{12} - \)\(34\!\cdots\!76\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 89 | \( 1 - \)\(15\!\cdots\!60\)\( p T^{2} + \)\(10\!\cdots\!84\)\( T^{4} - \)\(79\!\cdots\!20\)\( T^{6} + \)\(58\!\cdots\!86\)\( T^{8} - \)\(79\!\cdots\!20\)\( p^{48} T^{10} + \)\(10\!\cdots\!84\)\( p^{96} T^{12} - \)\(15\!\cdots\!60\)\( p^{145} T^{14} + p^{192} T^{16} \) | |
| 97 | \( ( 1 - \)\(40\!\cdots\!40\)\( T + \)\(12\!\cdots\!76\)\( T^{2} - \)\(73\!\cdots\!00\)\( T^{3} + \)\(76\!\cdots\!46\)\( T^{4} - \)\(73\!\cdots\!00\)\( p^{24} T^{5} + \)\(12\!\cdots\!76\)\( p^{48} T^{6} - \)\(40\!\cdots\!40\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
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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
−3.67305587964339001914624014351, −3.64915185698601655842630766009, −3.39398271351186455444275739897, −3.35214261794237376383883971292, −3.24385054791430515753068948248, −2.81739129759206069714673574120, −2.77501795465280630056773067351, −2.76263738145255698869376554296, −2.63773778886861712571719050481, −2.45421723702775386213338853991, −2.13875778409714647913744707543, −1.78663963224908624552469841819, −1.71897674192431920201505387069, −1.66820436529028736727679068003, −1.66024677699621693779982042617, −1.42174331039766793470849321307, −1.34389862486389762774753965612, −1.28554146366938594605158066695, −1.21387857195471855848124061284, −0.74170723258236286961662335591, −0.72262708069637397947083050219, −0.41448621220614708679735584858, −0.38964081166790271834485793612, −0.33258003434375767280224820878, −0.32694692915892374441726458299, 0.32694692915892374441726458299, 0.33258003434375767280224820878, 0.38964081166790271834485793612, 0.41448621220614708679735584858, 0.72262708069637397947083050219, 0.74170723258236286961662335591, 1.21387857195471855848124061284, 1.28554146366938594605158066695, 1.34389862486389762774753965612, 1.42174331039766793470849321307, 1.66024677699621693779982042617, 1.66820436529028736727679068003, 1.71897674192431920201505387069, 1.78663963224908624552469841819, 2.13875778409714647913744707543, 2.45421723702775386213338853991, 2.63773778886861712571719050481, 2.76263738145255698869376554296, 2.77501795465280630056773067351, 2.81739129759206069714673574120, 3.24385054791430515753068948248, 3.35214261794237376383883971292, 3.39398271351186455444275739897, 3.64915185698601655842630766009, 3.67305587964339001914624014351
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.