Properties

Label 16-4032e8-1.1-c1e8-0-4
Degree $16$
Conductor $6.985\times 10^{28}$
Sign $1$
Analytic cond. $1.15446\times 10^{12}$
Root an. cond. $5.67412$
Motivic weight $1$
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  − 8·7-s + 16·31-s + 36·49-s + 48·73-s + 32·79-s + 16·97-s − 16·103-s + 48·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s − 128·217-s + ⋯
L(s)  = 1  − 3.02·7-s + 2.87·31-s + 36/7·49-s + 5.61·73-s + 3.60·79-s + 1.62·97-s − 1.57·103-s + 4.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s − 8.68·217-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.15446\times 10^{12}\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6168155162\)
\(L(\frac12)\) \(\approx\) \(0.6168155162\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( ( 1 + T )^{8} \)
good5 \( ( 1 + 2 T^{4} + p^{4} T^{8} )^{2} \)
11 \( ( 1 - 24 T^{2} + 338 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 + 24 T^{2} + 290 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \)
23 \( ( 1 + 32 T^{2} + 882 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
31 \( ( 1 - 2 T + p T^{2} )^{8} \)
37 \( ( 1 + 4 T^{2} - 330 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
41 \( ( 1 + 120 T^{2} + 6530 T^{4} + 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 68 T^{2} + 4086 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 60 T^{2} + 4550 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - p T^{2} )^{8} \)
59 \( ( 1 - 108 T^{2} + 9110 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 212 T^{2} + 18486 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \)
71 \( ( 1 + 96 T^{2} + 12338 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \)
83 \( ( 1 - 92 T^{2} + 8982 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 120 T^{2} + 5570 T^{4} + 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \)
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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

−3.57217230772798702799643327765, −3.29975590125174843288839492627, −3.17286381945705356346708129112, −3.10840701572559444251812564095, −3.05291065991049613382503107102, −2.87697583361904601569520022758, −2.84433144646388543610649909206, −2.75485438197091057542766962252, −2.69958712860969712981486533592, −2.43055301078170922413407665175, −2.28050960033743704528443651185, −2.06296110267561332771424993829, −2.04960046875732945874460828531, −1.95535487342731189873950747168, −1.91670619290867144160380309885, −1.84803430663174840270228634703, −1.60108530836393839613214739820, −1.14377382191882001436251991719, −1.08729708573178246466219553686, −0.917722661272094446502291385769, −0.71701773634126550084430128492, −0.71173393842732793054112176960, −0.64145911297784914380168048625, −0.49396439431267312413863597662, −0.05416926191443243430912229401, 0.05416926191443243430912229401, 0.49396439431267312413863597662, 0.64145911297784914380168048625, 0.71173393842732793054112176960, 0.71701773634126550084430128492, 0.917722661272094446502291385769, 1.08729708573178246466219553686, 1.14377382191882001436251991719, 1.60108530836393839613214739820, 1.84803430663174840270228634703, 1.91670619290867144160380309885, 1.95535487342731189873950747168, 2.04960046875732945874460828531, 2.06296110267561332771424993829, 2.28050960033743704528443651185, 2.43055301078170922413407665175, 2.69958712860969712981486533592, 2.75485438197091057542766962252, 2.84433144646388543610649909206, 2.87697583361904601569520022758, 3.05291065991049613382503107102, 3.10840701572559444251812564095, 3.17286381945705356346708129112, 3.29975590125174843288839492627, 3.57217230772798702799643327765

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

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