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Properties

Label 16-3744e8-1.1-c0e8-0-0
Degree $16$
Conductor $3.861\times 10^{28}$
Sign $1$
Analytic cond. $148.574$
Root an. cond. $1.36693$
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  − 4·37-s + 4·49-s + 4·61-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 4·37-s + 4·49-s + 4·61-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 3^{16} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(148.574\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 3^{16} \cdot 13^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.028858547\)
\(L(\frac12)\) \(\approx\) \(2.028858547\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−3.96783930042591779735696690456, −3.48571749892104972999427417309, −3.41716736308766273314432241152, −3.34022161404617360440260588656, −3.30671932163189043116827413005, −3.25919914636652871795207376549, −3.13787001713402946857061454896, −2.84820791247217315699701265109, −2.71814478219887704315846168782, −2.66536395542221955841798533751, −2.59401780935714606018426831648, −2.42418731151182546518133533026, −2.37679593845727478132602435190, −2.07474879570368458487192460026, −2.03681433760597092425321884041, −1.95290697473395331420807836470, −1.78650493047866231281346873656, −1.68842767772875043835417660200, −1.62435642125160202309411944685, −1.29622903182375663117786838676, −1.13040613739690936279843104610, −1.00604317038343224894192755749, −0.841172754594099830753392102994, −0.61364463067819719144543609907, −0.39601022560456549636334922861, 0.39601022560456549636334922861, 0.61364463067819719144543609907, 0.841172754594099830753392102994, 1.00604317038343224894192755749, 1.13040613739690936279843104610, 1.29622903182375663117786838676, 1.62435642125160202309411944685, 1.68842767772875043835417660200, 1.78650493047866231281346873656, 1.95290697473395331420807836470, 2.03681433760597092425321884041, 2.07474879570368458487192460026, 2.37679593845727478132602435190, 2.42418731151182546518133533026, 2.59401780935714606018426831648, 2.66536395542221955841798533751, 2.71814478219887704315846168782, 2.84820791247217315699701265109, 3.13787001713402946857061454896, 3.25919914636652871795207376549, 3.30671932163189043116827413005, 3.34022161404617360440260588656, 3.41716736308766273314432241152, 3.48571749892104972999427417309, 3.96783930042591779735696690456

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

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