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Properties

Label 16-3528e8-1.1-c0e8-0-1
Degree $16$
Conductor $2.400\times 10^{28}$
Sign $1$
Analytic cond. $92.3603$
Root an. cond. $1.32691$
Motivic weight $0$
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  + 4·2-s + 6·4-s − 15·16-s − 4·25-s − 24·32-s − 16·50-s − 6·64-s − 4·67-s + 81-s − 24·100-s − 8·107-s − 2·121-s + 127-s + 36·128-s + 131-s − 16·134-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4·162-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 4·2-s + 6·4-s − 15·16-s − 4·25-s − 24·32-s − 16·50-s − 6·64-s − 4·67-s + 81-s − 24·100-s − 8·107-s − 2·121-s + 127-s + 36·128-s + 131-s − 16·134-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4·162-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 3^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(92.3603\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4197381545\)
\(L(\frac12)\) \(\approx\) \(0.4197381545\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} )^{4} \)
3 \( 1 - T^{4} + T^{8} \)
7 \( 1 \)
good5 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
11 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
13 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
17 \( ( 1 - T^{4} + T^{8} )^{2} \)
19 \( ( 1 - T^{4} + T^{8} )^{2} \)
23 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
29 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
31 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
37 \( ( 1 - T )^{8}( 1 + T )^{8} \)
41 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
43 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
47 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
53 \( ( 1 - T )^{8}( 1 + T )^{8} \)
59 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
61 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
67 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
71 \( ( 1 - T )^{8}( 1 + T )^{8} \)
73 \( ( 1 - T^{4} + T^{8} )^{2} \)
79 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
83 \( ( 1 - T^{4} + T^{8} )^{2} \)
89 \( ( 1 + T^{4} )^{4} \)
97 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
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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

−3.84826288297800173923075199762, −3.80366366789365053726931772513, −3.66489657535822245875869029983, −3.34051148185201151701433780914, −3.32417106842084341689523718941, −3.24563105715642110575072738644, −3.23035895290954522824569615620, −3.11193142689553640023757721636, −2.91422251177004733047000069810, −2.74454526374442050135675146608, −2.72310044057146837383441963864, −2.64609342594937052948965688691, −2.61014574952476767447689799275, −2.33971004068772555019600452656, −2.20021965301056666557740340748, −2.04133601313088608064545003158, −1.88767253160967516374031966888, −1.69295068886782093986890929912, −1.66440406824647333843646971451, −1.58311241627985061897308659216, −1.35878513977873447845626586687, −1.01237011341648419715215610516, −0.69972614601729358304220139664, −0.64000086400745112209743176605, −0.094650221076690233108040224055, 0.094650221076690233108040224055, 0.64000086400745112209743176605, 0.69972614601729358304220139664, 1.01237011341648419715215610516, 1.35878513977873447845626586687, 1.58311241627985061897308659216, 1.66440406824647333843646971451, 1.69295068886782093986890929912, 1.88767253160967516374031966888, 2.04133601313088608064545003158, 2.20021965301056666557740340748, 2.33971004068772555019600452656, 2.61014574952476767447689799275, 2.64609342594937052948965688691, 2.72310044057146837383441963864, 2.74454526374442050135675146608, 2.91422251177004733047000069810, 3.11193142689553640023757721636, 3.23035895290954522824569615620, 3.24563105715642110575072738644, 3.32417106842084341689523718941, 3.34051148185201151701433780914, 3.66489657535822245875869029983, 3.80366366789365053726931772513, 3.84826288297800173923075199762

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

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