Properties

Label 16-48e8-1.1-c22e8-0-1
Degree $16$
Conductor $2.818\times 10^{13}$
Sign $1$
Analytic cond. $2.20660\times 10^{17}$
Root an. cond. $12.1334$
Motivic weight $22$
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  + 5.26e7·5-s − 4.18e10·9-s − 1.61e12·13-s − 6.21e13·17-s − 3.20e15·25-s + 1.51e16·29-s − 3.12e17·37-s − 1.29e18·41-s − 2.20e18·45-s + 7.75e18·49-s − 2.08e19·53-s − 2.07e20·61-s − 8.48e19·65-s − 6.91e20·73-s + 1.09e21·81-s − 3.27e21·85-s + 3.83e21·89-s + 1.83e22·97-s + 4.68e22·101-s − 5.65e22·109-s − 1.41e23·113-s + 6.74e22·117-s + 1.04e23·121-s − 3.19e23·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.07·5-s − 4/3·9-s − 0.899·13-s − 1.81·17-s − 1.34·25-s + 1.24·29-s − 1.75·37-s − 2.35·41-s − 1.43·45-s + 1.98·49-s − 2.24·53-s − 4.77·61-s − 0.969·65-s − 2.20·73-s + 10/9·81-s − 1.95·85-s + 1.38·89-s + 2.56·97-s + 4.19·101-s − 2.19·109-s − 3.69·113-s + 1.19·117-s + 1.28·121-s − 2.74·125-s + 1.34·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(23-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+11)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(2.20660\times 10^{17}\)
Root analytic conductor: \(12.1334\)
Motivic weight: \(22\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{8} ,\ ( \ : [11]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{23}{2})\) \(\approx\) \(0.06557655676\)
\(L(\frac12)\) \(\approx\) \(0.06557655676\)
\(L(12)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + p^{21} T^{2} )^{4} \)
good5 \( ( 1 - 1053432 p^{2} T + 528171896744204 p T^{2} - 19946648865978351432 p^{4} T^{3} + \)\(75\!\cdots\!82\)\( p^{6} T^{4} - 19946648865978351432 p^{26} T^{5} + 528171896744204 p^{45} T^{6} - 1053432 p^{68} T^{7} + p^{88} T^{8} )^{2} \)
7 \( 1 - 7752742102538947784 T^{2} + \)\(12\!\cdots\!40\)\( p^{2} T^{4} - \)\(18\!\cdots\!88\)\( p^{5} T^{6} + \)\(23\!\cdots\!34\)\( p^{8} T^{8} - \)\(18\!\cdots\!88\)\( p^{49} T^{10} + \)\(12\!\cdots\!40\)\( p^{90} T^{12} - 7752742102538947784 p^{132} T^{14} + p^{176} T^{16} \)
11 \( 1 - \)\(95\!\cdots\!40\)\( p T^{2} + \)\(26\!\cdots\!44\)\( p^{2} T^{4} - \)\(19\!\cdots\!20\)\( p^{3} T^{6} + \)\(40\!\cdots\!66\)\( p^{4} T^{8} - \)\(19\!\cdots\!20\)\( p^{47} T^{10} + \)\(26\!\cdots\!44\)\( p^{90} T^{12} - \)\(95\!\cdots\!40\)\( p^{133} T^{14} + p^{176} T^{16} \)
13 \( ( 1 + 805732003640 T + \)\(60\!\cdots\!28\)\( T^{2} + \)\(87\!\cdots\!80\)\( p T^{3} + \)\(12\!\cdots\!42\)\( p^{2} T^{4} + \)\(87\!\cdots\!80\)\( p^{23} T^{5} + \)\(60\!\cdots\!28\)\( p^{44} T^{6} + 805732003640 p^{66} T^{7} + p^{88} T^{8} )^{2} \)
17 \( ( 1 + 31077123358680 T + \)\(15\!\cdots\!04\)\( p T^{2} + \)\(43\!\cdots\!40\)\( p^{2} T^{3} + \)\(50\!\cdots\!06\)\( p^{3} T^{4} + \)\(43\!\cdots\!40\)\( p^{24} T^{5} + \)\(15\!\cdots\!04\)\( p^{45} T^{6} + 31077123358680 p^{66} T^{7} + p^{88} T^{8} )^{2} \)
19 \( 1 - \)\(95\!\cdots\!76\)\( T^{2} + \)\(11\!\cdots\!60\)\( p^{2} T^{4} - \)\(80\!\cdots\!24\)\( p^{4} T^{6} + \)\(37\!\cdots\!74\)\( p^{6} T^{8} - \)\(80\!\cdots\!24\)\( p^{48} T^{10} + \)\(11\!\cdots\!60\)\( p^{90} T^{12} - \)\(95\!\cdots\!76\)\( p^{132} T^{14} + p^{176} T^{16} \)
23 \( 1 - \)\(61\!\cdots\!20\)\( T^{2} + \)\(74\!\cdots\!48\)\( p T^{4} - \)\(28\!\cdots\!60\)\( T^{6} + \)\(31\!\cdots\!66\)\( T^{8} - \)\(28\!\cdots\!60\)\( p^{44} T^{10} + \)\(74\!\cdots\!48\)\( p^{89} T^{12} - \)\(61\!\cdots\!20\)\( p^{132} T^{14} + p^{176} T^{16} \)
29 \( ( 1 - 7579892863668312 T + \)\(39\!\cdots\!28\)\( T^{2} - \)\(19\!\cdots\!44\)\( T^{3} + \)\(72\!\cdots\!30\)\( T^{4} - \)\(19\!\cdots\!44\)\( p^{22} T^{5} + \)\(39\!\cdots\!28\)\( p^{44} T^{6} - 7579892863668312 p^{66} T^{7} + p^{88} T^{8} )^{2} \)
31 \( 1 - \)\(27\!\cdots\!48\)\( T^{2} + \)\(42\!\cdots\!48\)\( T^{4} - \)\(44\!\cdots\!36\)\( T^{6} + \)\(33\!\cdots\!70\)\( T^{8} - \)\(44\!\cdots\!36\)\( p^{44} T^{10} + \)\(42\!\cdots\!48\)\( p^{88} T^{12} - \)\(27\!\cdots\!48\)\( p^{132} T^{14} + p^{176} T^{16} \)
37 \( ( 1 + 156320176691811160 T + \)\(92\!\cdots\!16\)\( T^{2} + \)\(10\!\cdots\!20\)\( T^{3} + \)\(39\!\cdots\!86\)\( T^{4} + \)\(10\!\cdots\!20\)\( p^{22} T^{5} + \)\(92\!\cdots\!16\)\( p^{44} T^{6} + 156320176691811160 p^{66} T^{7} + p^{88} T^{8} )^{2} \)
41 \( ( 1 + 648349497866384280 T + \)\(67\!\cdots\!24\)\( T^{2} + \)\(41\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!66\)\( T^{4} + \)\(41\!\cdots\!40\)\( p^{22} T^{5} + \)\(67\!\cdots\!24\)\( p^{44} T^{6} + 648349497866384280 p^{66} T^{7} + p^{88} T^{8} )^{2} \)
43 \( 1 - \)\(20\!\cdots\!20\)\( T^{2} + \)\(25\!\cdots\!44\)\( T^{4} - \)\(27\!\cdots\!60\)\( T^{6} + \)\(27\!\cdots\!86\)\( T^{8} - \)\(27\!\cdots\!60\)\( p^{44} T^{10} + \)\(25\!\cdots\!44\)\( p^{88} T^{12} - \)\(20\!\cdots\!20\)\( p^{132} T^{14} + p^{176} T^{16} \)
47 \( 1 - \)\(35\!\cdots\!04\)\( T^{2} + \)\(61\!\cdots\!20\)\( T^{4} - \)\(66\!\cdots\!56\)\( T^{6} + \)\(48\!\cdots\!34\)\( T^{8} - \)\(66\!\cdots\!56\)\( p^{44} T^{10} + \)\(61\!\cdots\!20\)\( p^{88} T^{12} - \)\(35\!\cdots\!04\)\( p^{132} T^{14} + p^{176} T^{16} \)
53 \( ( 1 + 10418254886440725480 T + \)\(28\!\cdots\!04\)\( T^{2} + \)\(22\!\cdots\!40\)\( T^{3} + \)\(35\!\cdots\!46\)\( T^{4} + \)\(22\!\cdots\!40\)\( p^{22} T^{5} + \)\(28\!\cdots\!04\)\( p^{44} T^{6} + 10418254886440725480 p^{66} T^{7} + p^{88} T^{8} )^{2} \)
59 \( 1 - \)\(47\!\cdots\!20\)\( T^{2} + \)\(11\!\cdots\!04\)\( T^{4} - \)\(17\!\cdots\!60\)\( T^{6} + \)\(18\!\cdots\!46\)\( T^{8} - \)\(17\!\cdots\!60\)\( p^{44} T^{10} + \)\(11\!\cdots\!04\)\( p^{88} T^{12} - \)\(47\!\cdots\!20\)\( p^{132} T^{14} + p^{176} T^{16} \)
61 \( ( 1 + \)\(10\!\cdots\!64\)\( T + \)\(93\!\cdots\!80\)\( T^{2} + \)\(57\!\cdots\!36\)\( T^{3} + \)\(28\!\cdots\!74\)\( T^{4} + \)\(57\!\cdots\!36\)\( p^{22} T^{5} + \)\(93\!\cdots\!80\)\( p^{44} T^{6} + \)\(10\!\cdots\!64\)\( p^{66} T^{7} + p^{88} T^{8} )^{2} \)
67 \( 1 - \)\(63\!\cdots\!24\)\( T^{2} + \)\(23\!\cdots\!60\)\( T^{4} - \)\(55\!\cdots\!96\)\( T^{6} + \)\(96\!\cdots\!94\)\( T^{8} - \)\(55\!\cdots\!96\)\( p^{44} T^{10} + \)\(23\!\cdots\!60\)\( p^{88} T^{12} - \)\(63\!\cdots\!24\)\( p^{132} T^{14} + p^{176} T^{16} \)
71 \( 1 - \)\(15\!\cdots\!00\)\( T^{2} + \)\(32\!\cdots\!64\)\( T^{4} - \)\(28\!\cdots\!00\)\( T^{6} + \)\(36\!\cdots\!46\)\( T^{8} - \)\(28\!\cdots\!00\)\( p^{44} T^{10} + \)\(32\!\cdots\!64\)\( p^{88} T^{12} - \)\(15\!\cdots\!00\)\( p^{132} T^{14} + p^{176} T^{16} \)
73 \( ( 1 + \)\(34\!\cdots\!80\)\( T + \)\(40\!\cdots\!08\)\( T^{2} + \)\(99\!\cdots\!60\)\( T^{3} + \)\(61\!\cdots\!18\)\( T^{4} + \)\(99\!\cdots\!60\)\( p^{22} T^{5} + \)\(40\!\cdots\!08\)\( p^{44} T^{6} + \)\(34\!\cdots\!80\)\( p^{66} T^{7} + p^{88} T^{8} )^{2} \)
79 \( 1 - \)\(27\!\cdots\!96\)\( T^{2} + \)\(39\!\cdots\!40\)\( T^{4} - \)\(36\!\cdots\!04\)\( T^{6} + \)\(24\!\cdots\!94\)\( T^{8} - \)\(36\!\cdots\!04\)\( p^{44} T^{10} + \)\(39\!\cdots\!40\)\( p^{88} T^{12} - \)\(27\!\cdots\!96\)\( p^{132} T^{14} + p^{176} T^{16} \)
83 \( 1 - \)\(58\!\cdots\!40\)\( T^{2} + \)\(16\!\cdots\!24\)\( T^{4} - \)\(30\!\cdots\!20\)\( T^{6} + \)\(51\!\cdots\!26\)\( T^{8} - \)\(30\!\cdots\!20\)\( p^{44} T^{10} + \)\(16\!\cdots\!24\)\( p^{88} T^{12} - \)\(58\!\cdots\!40\)\( p^{132} T^{14} + p^{176} T^{16} \)
89 \( ( 1 - \)\(19\!\cdots\!64\)\( T + \)\(10\!\cdots\!60\)\( T^{2} - \)\(27\!\cdots\!96\)\( T^{3} + \)\(41\!\cdots\!14\)\( T^{4} - \)\(27\!\cdots\!96\)\( p^{22} T^{5} + \)\(10\!\cdots\!60\)\( p^{44} T^{6} - \)\(19\!\cdots\!64\)\( p^{66} T^{7} + p^{88} T^{8} )^{2} \)
97 \( ( 1 - \)\(91\!\cdots\!00\)\( T + \)\(39\!\cdots\!44\)\( T^{2} - \)\(47\!\cdots\!00\)\( T^{3} + \)\(57\!\cdots\!46\)\( T^{4} - \)\(47\!\cdots\!00\)\( p^{22} T^{5} + \)\(39\!\cdots\!44\)\( p^{44} T^{6} - \)\(91\!\cdots\!00\)\( p^{66} T^{7} + p^{88} T^{8} )^{2} \)
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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

−3.87371320255782618112280878466, −3.49686588060894703465912525428, −3.37905088154538601537378070526, −3.36701820139153959003300066938, −3.26160232933989737779088332746, −3.09401832564176272487293873266, −2.83026753244480637769739293654, −2.73076115487997247151392724781, −2.55900338421495759032373236550, −2.33371339822476434137017363332, −2.30054750335652066808625967727, −2.05906500270560630799117865613, −1.99785571159299574667059270617, −1.84765367251605005222974578151, −1.81051954623599720485661989443, −1.46123642349236299275308047596, −1.34732044145364766250848891934, −1.22807799891253642104648433941, −1.17346159195495197546903968206, −0.857490832631475373295724867857, −0.69519755687180958389420777657, −0.37946360517893303839629255554, −0.29412819873994563644853977819, −0.10875815930719518069910596590, −0.05024598068864194678692806718, 0.05024598068864194678692806718, 0.10875815930719518069910596590, 0.29412819873994563644853977819, 0.37946360517893303839629255554, 0.69519755687180958389420777657, 0.857490832631475373295724867857, 1.17346159195495197546903968206, 1.22807799891253642104648433941, 1.34732044145364766250848891934, 1.46123642349236299275308047596, 1.81051954623599720485661989443, 1.84765367251605005222974578151, 1.99785571159299574667059270617, 2.05906500270560630799117865613, 2.30054750335652066808625967727, 2.33371339822476434137017363332, 2.55900338421495759032373236550, 2.73076115487997247151392724781, 2.83026753244480637769739293654, 3.09401832564176272487293873266, 3.26160232933989737779088332746, 3.36701820139153959003300066938, 3.37905088154538601537378070526, 3.49686588060894703465912525428, 3.87371320255782618112280878466

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

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