Properties

Label 16-6e8-1.1-c24e8-0-0
Degree $16$
Conductor $1679616$
Sign $1$
Analytic cond. $5.28737\times 10^{10}$
Root an. cond. $4.67953$
Motivic weight $24$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.31e5·3-s − 3.35e7·4-s − 1.01e10·7-s + 1.56e11·9-s + 4.42e12·12-s + 5.05e13·13-s + 7.03e14·16-s − 9.78e14·19-s + 1.34e15·21-s + 2.42e17·25-s − 4.87e16·27-s + 3.40e17·28-s + 3.09e18·31-s − 5.24e18·36-s + 2.25e19·37-s − 6.66e18·39-s + 2.26e20·43-s − 9.28e19·48-s − 6.62e20·49-s − 1.69e21·52-s + 1.28e20·57-s − 6.50e21·61-s − 1.58e21·63-s − 1.18e22·64-s − 7.04e21·67-s + 3.62e22·73-s − 3.19e22·75-s + ⋯
L(s)  = 1  − 0.248·3-s − 2·4-s − 0.734·7-s + 0.553·9-s + 0.496·12-s + 2.17·13-s + 5/2·16-s − 0.441·19-s + 0.182·21-s + 4.06·25-s − 0.324·27-s + 1.46·28-s + 3.92·31-s − 1.10·36-s + 3.43·37-s − 0.538·39-s + 5.66·43-s − 0.620·48-s − 3.45·49-s − 4.34·52-s + 0.109·57-s − 2.45·61-s − 0.406·63-s − 5/2·64-s − 0.860·67-s + 1.58·73-s − 1.00·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+12)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(1679616\)    =    \(2^{8} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(5.28737\times 10^{10}\)
Root analytic conductor: \(4.67953\)
Motivic weight: \(24\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 1679616,\ (\ :[12]^{8}),\ 1)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(27.95206882\)
\(L(\frac12)\) \(\approx\) \(27.95206882\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p^{23} T^{2} )^{4} \)
3 \( 1 + 43960 p T - 190519012 p^{6} T^{2} + 6136572680 p^{13} T^{3} + 2567030625142 p^{22} T^{4} + 6136572680 p^{37} T^{5} - 190519012 p^{54} T^{6} + 43960 p^{73} T^{7} + p^{96} T^{8} \)
good5 \( 1 - 242233888485069512 T^{2} + \)\(16\!\cdots\!76\)\( p^{3} T^{4} - \)\(42\!\cdots\!12\)\( p^{6} T^{6} + \)\(89\!\cdots\!26\)\( p^{14} T^{8} - \)\(42\!\cdots\!12\)\( p^{54} T^{10} + \)\(16\!\cdots\!76\)\( p^{99} T^{12} - 242233888485069512 p^{144} T^{14} + p^{192} T^{16} \)
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

−6.13435039509716438315737053670, −5.94896674554765800587462846112, −5.58259360166925018181375303861, −5.43916590617675305517866381095, −4.77829198380849522749416059564, −4.77507282208977231312617984814, −4.51683358311718038156666834832, −4.41935538334447090486624688009, −4.39453258195576504780431382451, −4.13828093260468041706972791137, −3.61993267637823713923120818443, −3.32089522526169056143070494622, −3.19496260735458942556410499950, −3.06021780283389970378503335732, −2.70409848570300123722015936925, −2.65544190318909419815519909205, −2.12495071391108041714130029764, −1.82589380788275317168779895448, −1.49601786135068435117804322497, −1.18930130381941536553543523570, −0.76899749529298679323563685936, −0.74175779133109537033970377374, −0.63596541731637220328247265360, −0.61567017855446826661169655473, −0.61232541560928236478977761919, 0.61232541560928236478977761919, 0.61567017855446826661169655473, 0.63596541731637220328247265360, 0.74175779133109537033970377374, 0.76899749529298679323563685936, 1.18930130381941536553543523570, 1.49601786135068435117804322497, 1.82589380788275317168779895448, 2.12495071391108041714130029764, 2.65544190318909419815519909205, 2.70409848570300123722015936925, 3.06021780283389970378503335732, 3.19496260735458942556410499950, 3.32089522526169056143070494622, 3.61993267637823713923120818443, 4.13828093260468041706972791137, 4.39453258195576504780431382451, 4.41935538334447090486624688009, 4.51683358311718038156666834832, 4.77507282208977231312617984814, 4.77829198380849522749416059564, 5.43916590617675305517866381095, 5.58259360166925018181375303861, 5.94896674554765800587462846112, 6.13435039509716438315737053670

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.