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Properties

Label 16-329e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.373\times 10^{20}$
Sign $1$
Analytic cond. $5.28231\times 10^{-7}$
Root an. cond. $0.405206$
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  − 2-s + 2·3-s − 2·6-s + 7-s − 8-s + 3·9-s − 14-s + 16-s − 17-s − 3·18-s + 2·21-s − 2·24-s − 4·25-s + 2·27-s + 32-s + 34-s − 37-s − 2·42-s − 4·47-s + 2·48-s + 49-s + 4·50-s − 2·51-s − 53-s − 2·54-s − 56-s − 59-s + ⋯
L(s)  = 1  − 2-s + 2·3-s − 2·6-s + 7-s − 8-s + 3·9-s − 14-s + 16-s − 17-s − 3·18-s + 2·21-s − 2·24-s − 4·25-s + 2·27-s + 32-s + 34-s − 37-s − 2·42-s − 4·47-s + 2·48-s + 49-s + 4·50-s − 2·51-s − 53-s − 2·54-s − 56-s − 59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 47^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 47^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(7^{8} \cdot 47^{8}\)
Sign: $1$
Analytic conductor: \(5.28231\times 10^{-7}\)
Root analytic conductor: \(0.405206\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 7^{8} \cdot 47^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1527470971\)
\(L(\frac12)\) \(\approx\) \(0.1527470971\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \)
47 \( ( 1 + T + T^{2} )^{4} \)
good2 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
3 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
5 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
11 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
13 \( ( 1 - T )^{8}( 1 + T )^{8} \)
17 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
19 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
23 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
29 \( ( 1 - T )^{8}( 1 + T )^{8} \)
31 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
37 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
41 \( ( 1 - T )^{8}( 1 + T )^{8} \)
43 \( ( 1 - T )^{8}( 1 + T )^{8} \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
59 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
61 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
67 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
71 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
73 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
79 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
83 \( ( 1 + T + T^{2} )^{8} \)
89 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
97 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
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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

−5.47907303025714143626134814770, −5.18118346552035789536329621453, −5.13171704548076107549426400459, −5.06753779487272236067716293054, −4.86023436445515047227807239936, −4.76386455813186233448959683750, −4.31942202782266471653121033957, −4.31284991009843045661816457700, −4.30098233821970231797049191624, −4.07457665983559261799411358079, −3.86654237710398461413584328865, −3.85318803289625649839914512541, −3.64692453493006510428542000412, −3.35300331510020270218540063404, −3.30804352399552803289161103522, −2.98684283668355827670913848217, −2.90183104402826162231304474382, −2.86233327996632195942287932181, −2.41612396739851796074505688589, −2.26449363185006760463762289466, −2.11133047432001115468590785707, −1.80868600520281066469065676046, −1.68645756168233095062941011887, −1.48246323250519140429948159420, −1.47036536131553938298971534235, 1.47036536131553938298971534235, 1.48246323250519140429948159420, 1.68645756168233095062941011887, 1.80868600520281066469065676046, 2.11133047432001115468590785707, 2.26449363185006760463762289466, 2.41612396739851796074505688589, 2.86233327996632195942287932181, 2.90183104402826162231304474382, 2.98684283668355827670913848217, 3.30804352399552803289161103522, 3.35300331510020270218540063404, 3.64692453493006510428542000412, 3.85318803289625649839914512541, 3.86654237710398461413584328865, 4.07457665983559261799411358079, 4.30098233821970231797049191624, 4.31284991009843045661816457700, 4.31942202782266471653121033957, 4.76386455813186233448959683750, 4.86023436445515047227807239936, 5.06753779487272236067716293054, 5.13171704548076107549426400459, 5.18118346552035789536329621453, 5.47907303025714143626134814770

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

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