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Properties

Label 16-3920e8-1.1-c0e8-0-1
Degree $16$
Conductor $5.576\times 10^{28}$
Sign $1$
Analytic cond. $214.558$
Root an. cond. $1.39869$
Motivic weight $0$
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  − 4·11-s − 4·23-s + 4·37-s + 8·43-s − 81-s + 4·107-s − 8·113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 4·11-s − 4·23-s + 4·37-s + 8·43-s − 81-s + 4·107-s − 8·113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 5^{8} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(214.558\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 5^{8} \cdot 7^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.784939724\)
\(L(\frac12)\) \(\approx\) \(1.784939724\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T^{4} + T^{8} \)
7 \( 1 \)
good3 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
11 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
13 \( ( 1 - T^{4} + T^{8} )^{2} \)
17 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
19 \( ( 1 - T^{4} + T^{8} )^{2} \)
23 \( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
29 \( ( 1 - T^{2} + T^{4} )^{4} \)
31 \( ( 1 - T^{2} + T^{4} )^{4} \)
37 \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
41 \( ( 1 + T^{2} )^{8} \)
43 \( ( 1 - T )^{8}( 1 + T^{2} )^{4} \)
47 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
53 \( ( 1 - T^{4} + T^{8} )^{2} \)
59 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
61 \( ( 1 - T^{2} + T^{4} )^{4} \)
67 \( ( 1 - T^{4} + T^{8} )^{2} \)
71 \( ( 1 + T^{2} )^{8} \)
73 \( ( 1 - T^{4} + T^{8} )^{2} \)
79 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
83 \( ( 1 + T^{4} )^{4} \)
89 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
97 \( ( 1 - T^{4} + T^{8} )^{2} \)
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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

−3.70247410072498112661578133924, −3.68228156418811431299079819086, −3.61551579642269161278477767307, −3.33513936163986294472662268923, −3.09595531842012476603387382358, −3.04424797959268466719976104693, −2.80564254438254997010884458091, −2.79682678977230978284815060116, −2.73868125658618933560953525388, −2.56001020492088776395240421557, −2.55707034508555427875102035174, −2.45882652140681724943220499133, −2.44965295178902116507691057748, −2.41705656978140275097639512051, −2.00114286070444503576532494219, −1.97681050069439424252177274782, −1.70189108798486642703206418787, −1.66612188616610596524962516862, −1.65664357809761840479719410472, −1.48481874723939200215319785259, −0.927512007171703034816921796165, −0.904380664788448816773588719542, −0.62656776523311349125675141608, −0.62641060186266334874102571313, −0.50100356431413836035387006312, 0.50100356431413836035387006312, 0.62641060186266334874102571313, 0.62656776523311349125675141608, 0.904380664788448816773588719542, 0.927512007171703034816921796165, 1.48481874723939200215319785259, 1.65664357809761840479719410472, 1.66612188616610596524962516862, 1.70189108798486642703206418787, 1.97681050069439424252177274782, 2.00114286070444503576532494219, 2.41705656978140275097639512051, 2.44965295178902116507691057748, 2.45882652140681724943220499133, 2.55707034508555427875102035174, 2.56001020492088776395240421557, 2.73868125658618933560953525388, 2.79682678977230978284815060116, 2.80564254438254997010884458091, 3.04424797959268466719976104693, 3.09595531842012476603387382358, 3.33513936163986294472662268923, 3.61551579642269161278477767307, 3.68228156418811431299079819086, 3.70247410072498112661578133924

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

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