Properties

Label 16-48e8-1.1-c24e8-0-2
Degree $16$
Conductor $2.818\times 10^{13}$
Sign $1$
Analytic cond. $8.87074\times 10^{17}$
Root an. cond. $13.2357$
Motivic weight $24$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.00e8·5-s − 3.76e11·9-s − 2.12e13·13-s + 8.31e14·17-s − 1.94e17·25-s + 3.56e17·29-s + 1.36e19·37-s − 3.05e19·41-s + 3.79e19·45-s + 8.66e20·49-s + 1.12e21·53-s + 1.15e22·61-s + 2.14e21·65-s + 1.04e23·73-s + 8.86e22·81-s − 8.37e22·85-s + 1.11e23·89-s − 2.49e24·97-s − 4.76e24·101-s − 7.78e24·109-s − 8.99e24·113-s + 8.01e24·117-s + 3.52e25·121-s + 3.64e25·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.412·5-s − 4/3·9-s − 0.913·13-s + 1.42·17-s − 3.26·25-s + 1.00·29-s + 2.07·37-s − 1.35·41-s + 0.550·45-s + 4.52·49-s + 2.28·53-s + 4.35·61-s + 0.376·65-s + 4.57·73-s + 10/9·81-s − 0.588·85-s + 0.452·89-s − 3.59·97-s − 4.22·101-s − 2.76·109-s − 2.07·113-s + 1.21·117-s + 3.57·121-s + 2.50·125-s − 0.415·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(25-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+12)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

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

\(L(\frac{25}{2})\) \(\approx\) \(16.35554497\)
\(L(\frac12)\) \(\approx\) \(16.35554497\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + p^{23} T^{2} )^{4} \)
good5 \( ( 1 + 50355144 T + 809273249518028 p^{3} T^{2} - \)\(25\!\cdots\!44\)\( p^{3} T^{3} + \)\(90\!\cdots\!78\)\( p^{7} T^{4} - \)\(25\!\cdots\!44\)\( p^{27} T^{5} + 809273249518028 p^{51} T^{6} + 50355144 p^{72} T^{7} + p^{96} T^{8} )^{2} \)
7 \( 1 - \)\(86\!\cdots\!56\)\( T^{2} + \)\(83\!\cdots\!40\)\( p^{2} T^{4} - \)\(10\!\cdots\!16\)\( p^{6} T^{6} + \)\(41\!\cdots\!86\)\( p^{14} T^{8} - \)\(10\!\cdots\!16\)\( p^{54} T^{10} + \)\(83\!\cdots\!40\)\( p^{98} T^{12} - \)\(86\!\cdots\!56\)\( p^{144} T^{14} + p^{192} T^{16} \)
11 \( 1 - \)\(35\!\cdots\!20\)\( T^{2} + \)\(51\!\cdots\!24\)\( p^{2} T^{4} - \)\(50\!\cdots\!60\)\( p^{6} T^{6} + \)\(39\!\cdots\!26\)\( p^{10} T^{8} - \)\(50\!\cdots\!60\)\( p^{54} T^{10} + \)\(51\!\cdots\!24\)\( p^{98} T^{12} - \)\(35\!\cdots\!20\)\( p^{144} T^{14} + p^{192} T^{16} \)
13 \( ( 1 + 10638303323480 T + \)\(15\!\cdots\!32\)\( T^{2} + \)\(85\!\cdots\!00\)\( p T^{3} + \)\(67\!\cdots\!02\)\( p^{2} T^{4} + \)\(85\!\cdots\!00\)\( p^{25} T^{5} + \)\(15\!\cdots\!32\)\( p^{48} T^{6} + 10638303323480 p^{72} T^{7} + p^{96} T^{8} )^{2} \)
17 \( ( 1 - 415650238715880 T + \)\(26\!\cdots\!56\)\( p T^{2} - \)\(91\!\cdots\!00\)\( p^{2} T^{3} + \)\(45\!\cdots\!46\)\( p^{3} T^{4} - \)\(91\!\cdots\!00\)\( p^{26} T^{5} + \)\(26\!\cdots\!56\)\( p^{49} T^{6} - 415650238715880 p^{72} T^{7} + p^{96} T^{8} )^{2} \)
19 \( 1 - \)\(20\!\cdots\!96\)\( T^{2} + \)\(63\!\cdots\!00\)\( p^{2} T^{4} - \)\(13\!\cdots\!64\)\( p^{4} T^{6} + \)\(20\!\cdots\!74\)\( p^{6} T^{8} - \)\(13\!\cdots\!64\)\( p^{52} T^{10} + \)\(63\!\cdots\!00\)\( p^{98} T^{12} - \)\(20\!\cdots\!96\)\( p^{144} T^{14} + p^{192} T^{16} \)
23 \( 1 - \)\(48\!\cdots\!80\)\( p^{2} T^{2} + \)\(11\!\cdots\!84\)\( p^{4} T^{4} - \)\(16\!\cdots\!40\)\( p^{6} T^{6} + \)\(18\!\cdots\!26\)\( p^{8} T^{8} - \)\(16\!\cdots\!40\)\( p^{54} T^{10} + \)\(11\!\cdots\!84\)\( p^{100} T^{12} - \)\(48\!\cdots\!80\)\( p^{146} T^{14} + p^{192} T^{16} \)
29 \( ( 1 - 178275139034136408 T + \)\(94\!\cdots\!28\)\( T^{2} - \)\(16\!\cdots\!96\)\( T^{3} + \)\(27\!\cdots\!50\)\( T^{4} - \)\(16\!\cdots\!96\)\( p^{24} T^{5} + \)\(94\!\cdots\!28\)\( p^{48} T^{6} - 178275139034136408 p^{72} T^{7} + p^{96} T^{8} )^{2} \)
31 \( 1 - \)\(30\!\cdots\!48\)\( T^{2} + \)\(45\!\cdots\!28\)\( T^{4} - \)\(45\!\cdots\!16\)\( T^{6} + \)\(33\!\cdots\!70\)\( T^{8} - \)\(45\!\cdots\!16\)\( p^{48} T^{10} + \)\(45\!\cdots\!28\)\( p^{96} T^{12} - \)\(30\!\cdots\!48\)\( p^{144} T^{14} + p^{192} T^{16} \)
37 \( ( 1 - 6838804222493876360 T + \)\(64\!\cdots\!24\)\( T^{2} - \)\(53\!\cdots\!80\)\( T^{3} + \)\(50\!\cdots\!86\)\( T^{4} - \)\(53\!\cdots\!80\)\( p^{24} T^{5} + \)\(64\!\cdots\!24\)\( p^{48} T^{6} - 6838804222493876360 p^{72} T^{7} + p^{96} T^{8} )^{2} \)
41 \( ( 1 + 15286483899817343640 T + \)\(11\!\cdots\!44\)\( T^{2} + \)\(16\!\cdots\!20\)\( T^{3} + \)\(54\!\cdots\!26\)\( T^{4} + \)\(16\!\cdots\!20\)\( p^{24} T^{5} + \)\(11\!\cdots\!44\)\( p^{48} T^{6} + 15286483899817343640 p^{72} T^{7} + p^{96} T^{8} )^{2} \)
43 \( 1 - \)\(41\!\cdots\!60\)\( T^{2} + \)\(95\!\cdots\!24\)\( T^{4} - \)\(16\!\cdots\!80\)\( T^{6} + \)\(26\!\cdots\!46\)\( T^{8} - \)\(16\!\cdots\!80\)\( p^{48} T^{10} + \)\(95\!\cdots\!24\)\( p^{96} T^{12} - \)\(41\!\cdots\!60\)\( p^{144} T^{14} + p^{192} T^{16} \)
47 \( 1 - \)\(40\!\cdots\!96\)\( T^{2} + \)\(11\!\cdots\!20\)\( T^{4} - \)\(20\!\cdots\!24\)\( T^{6} + \)\(30\!\cdots\!94\)\( T^{8} - \)\(20\!\cdots\!24\)\( p^{48} T^{10} + \)\(11\!\cdots\!20\)\( p^{96} T^{12} - \)\(40\!\cdots\!96\)\( p^{144} T^{14} + p^{192} T^{16} \)
53 \( ( 1 - \)\(56\!\cdots\!40\)\( T + \)\(88\!\cdots\!36\)\( T^{2} - \)\(35\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!86\)\( T^{4} - \)\(35\!\cdots\!40\)\( p^{24} T^{5} + \)\(88\!\cdots\!36\)\( p^{48} T^{6} - \)\(56\!\cdots\!40\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \)
59 \( 1 - \)\(12\!\cdots\!60\)\( T^{2} + \)\(62\!\cdots\!64\)\( T^{4} - \)\(20\!\cdots\!80\)\( T^{6} + \)\(62\!\cdots\!06\)\( T^{8} - \)\(20\!\cdots\!80\)\( p^{48} T^{10} + \)\(62\!\cdots\!64\)\( p^{96} T^{12} - \)\(12\!\cdots\!60\)\( p^{144} T^{14} + p^{192} T^{16} \)
61 \( ( 1 - \)\(57\!\cdots\!96\)\( T + \)\(30\!\cdots\!00\)\( T^{2} - \)\(10\!\cdots\!44\)\( T^{3} + \)\(32\!\cdots\!94\)\( T^{4} - \)\(10\!\cdots\!44\)\( p^{24} T^{5} + \)\(30\!\cdots\!00\)\( p^{48} T^{6} - \)\(57\!\cdots\!96\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \)
67 \( 1 - \)\(24\!\cdots\!56\)\( T^{2} + \)\(33\!\cdots\!20\)\( T^{4} - \)\(30\!\cdots\!04\)\( T^{6} + \)\(22\!\cdots\!74\)\( T^{8} - \)\(30\!\cdots\!04\)\( p^{48} T^{10} + \)\(33\!\cdots\!20\)\( p^{96} T^{12} - \)\(24\!\cdots\!56\)\( p^{144} T^{14} + p^{192} T^{16} \)
71 \( 1 - \)\(17\!\cdots\!60\)\( T^{2} + \)\(14\!\cdots\!64\)\( T^{4} - \)\(70\!\cdots\!80\)\( T^{6} + \)\(23\!\cdots\!66\)\( T^{8} - \)\(70\!\cdots\!80\)\( p^{48} T^{10} + \)\(14\!\cdots\!64\)\( p^{96} T^{12} - \)\(17\!\cdots\!60\)\( p^{144} T^{14} + p^{192} T^{16} \)
73 \( ( 1 - \)\(52\!\cdots\!40\)\( T + \)\(15\!\cdots\!12\)\( T^{2} - \)\(36\!\cdots\!40\)\( T^{3} + \)\(79\!\cdots\!58\)\( T^{4} - \)\(36\!\cdots\!40\)\( p^{24} T^{5} + \)\(15\!\cdots\!12\)\( p^{48} T^{6} - \)\(52\!\cdots\!40\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \)
79 \( 1 - \)\(11\!\cdots\!76\)\( T^{2} + \)\(52\!\cdots\!40\)\( T^{4} - \)\(15\!\cdots\!84\)\( T^{6} + \)\(43\!\cdots\!14\)\( T^{8} - \)\(15\!\cdots\!84\)\( p^{48} T^{10} + \)\(52\!\cdots\!40\)\( p^{96} T^{12} - \)\(11\!\cdots\!76\)\( p^{144} T^{14} + p^{192} T^{16} \)
83 \( 1 - \)\(52\!\cdots\!40\)\( T^{2} + \)\(15\!\cdots\!84\)\( T^{4} - \)\(28\!\cdots\!20\)\( T^{6} + \)\(38\!\cdots\!26\)\( T^{8} - \)\(28\!\cdots\!20\)\( p^{48} T^{10} + \)\(15\!\cdots\!84\)\( p^{96} T^{12} - \)\(52\!\cdots\!40\)\( p^{144} T^{14} + p^{192} T^{16} \)
89 \( ( 1 - \)\(55\!\cdots\!96\)\( T + \)\(23\!\cdots\!40\)\( T^{2} - \)\(10\!\cdots\!64\)\( T^{3} + \)\(21\!\cdots\!14\)\( T^{4} - \)\(10\!\cdots\!64\)\( p^{24} T^{5} + \)\(23\!\cdots\!40\)\( p^{48} T^{6} - \)\(55\!\cdots\!96\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \)
97 \( ( 1 + \)\(12\!\cdots\!00\)\( T + \)\(18\!\cdots\!96\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!26\)\( T^{4} + \)\(16\!\cdots\!00\)\( p^{24} T^{5} + \)\(18\!\cdots\!96\)\( p^{48} T^{6} + \)\(12\!\cdots\!00\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \)
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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

−3.69632396332597091709027854914, −3.59384577122821886912866806070, −3.57238176021569479070503648127, −3.41462178465546381653200144255, −3.04368380441830666602747108507, −2.83909252032767750982774446470, −2.67302334185155163369940588494, −2.50423666813127434570957832017, −2.50216860783828956212200716863, −2.40515763918533708155804312415, −2.29392322200112016763647524897, −2.23084691869109194273472374209, −2.10711754827179198445622392675, −1.65191817152422584955859434268, −1.45736359070236693599030253201, −1.41868642567913409303154152482, −1.40542501643205789399962458199, −1.08544057953998926039028763198, −0.904742278022106278250276085872, −0.72079081981163891878859545095, −0.68541204881796546679228094184, −0.51507500164472485278786080072, −0.38289090213833935629820995793, −0.32050700316930773237346207131, −0.19634440849323425892287087193, 0.19634440849323425892287087193, 0.32050700316930773237346207131, 0.38289090213833935629820995793, 0.51507500164472485278786080072, 0.68541204881796546679228094184, 0.72079081981163891878859545095, 0.904742278022106278250276085872, 1.08544057953998926039028763198, 1.40542501643205789399962458199, 1.41868642567913409303154152482, 1.45736359070236693599030253201, 1.65191817152422584955859434268, 2.10711754827179198445622392675, 2.23084691869109194273472374209, 2.29392322200112016763647524897, 2.40515763918533708155804312415, 2.50216860783828956212200716863, 2.50423666813127434570957832017, 2.67302334185155163369940588494, 2.83909252032767750982774446470, 3.04368380441830666602747108507, 3.41462178465546381653200144255, 3.57238176021569479070503648127, 3.59384577122821886912866806070, 3.69632396332597091709027854914

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

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