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Properties

Label 16-651e8-1.1-c0e8-0-1
Degree $16$
Conductor $3.226\times 10^{22}$
Sign $1$
Analytic cond. $0.000124138$
Root an. cond. $0.569992$
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  + 3-s + 4-s − 2·7-s + 9-s + 12-s − 13-s + 16-s − 4·19-s − 2·21-s − 4·25-s − 2·28-s − 2·31-s + 36-s + 2·37-s − 39-s − 3·43-s + 48-s + 49-s − 52-s − 4·57-s + 2·61-s − 2·63-s + 2·67-s + 2·73-s − 4·75-s − 4·76-s − 3·79-s + ⋯
L(s)  = 1  + 3-s + 4-s − 2·7-s + 9-s + 12-s − 13-s + 16-s − 4·19-s − 2·21-s − 4·25-s − 2·28-s − 2·31-s + 36-s + 2·37-s − 39-s − 3·43-s + 48-s + 49-s − 52-s − 4·57-s + 2·61-s − 2·63-s + 2·67-s + 2·73-s − 4·75-s − 4·76-s − 3·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 7^{8} \cdot 31^{8}\)
Sign: $1$
Analytic conductor: \(0.000124138\)
Root analytic conductor: \(0.569992\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 7^{8} \cdot 31^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2772176715\)
\(L(\frac12)\) \(\approx\) \(0.2772176715\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \)
7 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
good2 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
5 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
13 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
17 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
19 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
23 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
37 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
41 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
43 \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
47 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
59 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
61 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
67 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
71 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
73 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
79 \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
83 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
89 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
97 \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
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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

−4.84821202057512770109306802174, −4.74602667927920448723779925573, −4.43323154190223446830396392711, −4.30193905562669461628573406265, −4.17646645182787123042334246176, −4.06191663976126834718315124642, −3.99370605204720060172498121044, −3.99198053591766209026616235620, −3.71997412758502302553452477808, −3.54386394717633285480264322686, −3.42491901842239858302934522219, −3.28247010918197281618924462172, −3.24545865717958714391812649382, −3.17342897288664501476071486740, −2.68991406477638339028641626293, −2.46157365467604222398881785406, −2.40619244473964766150778881077, −2.28856779257757641076535811888, −2.27884065696770729030605928770, −2.25616189804726936060309226302, −1.70930171994548922380592155055, −1.69468057079276969905177756042, −1.59586756927977598551685901378, −1.35098527232529073809771819046, −0.45662934686396610803421945923, 0.45662934686396610803421945923, 1.35098527232529073809771819046, 1.59586756927977598551685901378, 1.69468057079276969905177756042, 1.70930171994548922380592155055, 2.25616189804726936060309226302, 2.27884065696770729030605928770, 2.28856779257757641076535811888, 2.40619244473964766150778881077, 2.46157365467604222398881785406, 2.68991406477638339028641626293, 3.17342897288664501476071486740, 3.24545865717958714391812649382, 3.28247010918197281618924462172, 3.42491901842239858302934522219, 3.54386394717633285480264322686, 3.71997412758502302553452477808, 3.99198053591766209026616235620, 3.99370605204720060172498121044, 4.06191663976126834718315124642, 4.17646645182787123042334246176, 4.30193905562669461628573406265, 4.43323154190223446830396392711, 4.74602667927920448723779925573, 4.84821202057512770109306802174

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

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