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Properties

Label 16-651e8-1.1-c0e8-0-2
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 − 2·21-s + 4·25-s + 2·28-s − 2·31-s + 36-s + 39-s − 5·43-s − 48-s + 49-s − 52-s − 2·61-s + 2·63-s − 2·67-s − 2·73-s − 4·75-s − 5·79-s − 2·84-s − 2·91-s + 2·93-s + 5·97-s + ⋯
L(s)  = 1  − 3-s + 4-s + 2·7-s + 9-s − 12-s − 13-s + 16-s − 2·21-s + 4·25-s + 2·28-s − 2·31-s + 36-s + 39-s − 5·43-s − 48-s + 49-s − 52-s − 2·61-s + 2·63-s − 2·67-s − 2·73-s − 4·75-s − 5·79-s − 2·84-s − 2·91-s + 2·93-s + 5·97-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

\(L(\frac{1}{2})\) \(\approx\) \(0.5359730869\)
\(L(\frac12)\) \(\approx\) \(0.5359730869\)
\(L(1)\) 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 \( ( 1 - T^{2} + T^{4} )^{4} \)
11 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
29 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
37 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
41 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
43 \( ( 1 + T + T^{2} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
47 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
59 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
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

−4.84674517702230403650068764380, −4.64252334459921917164349033753, −4.57872407996006423224708051654, −4.49941256492330002953770711382, −4.38940292218467137034842248072, −4.35778378484387692664984459778, −4.31017606298920287966247798879, −3.97406507175118712634486873192, −3.67258046559083068700654905130, −3.56542110206234700093253600260, −3.19739999332537623057432062075, −3.15304839180066403179923311610, −3.13624998703358259119022378553, −3.12542413203795366861439631168, −3.03577050843692710044128975195, −2.64315442897944884531906893966, −2.56364786863615259657585464143, −2.18584880663049809929505528333, −2.08706290873058233677056203202, −1.76343792973581510055546838657, −1.75431062844342791012193579223, −1.51162451742187009908975125735, −1.38778976535427244673997321585, −1.33045710702257110428968924352, −0.923737363406021098756849252173, 0.923737363406021098756849252173, 1.33045710702257110428968924352, 1.38778976535427244673997321585, 1.51162451742187009908975125735, 1.75431062844342791012193579223, 1.76343792973581510055546838657, 2.08706290873058233677056203202, 2.18584880663049809929505528333, 2.56364786863615259657585464143, 2.64315442897944884531906893966, 3.03577050843692710044128975195, 3.12542413203795366861439631168, 3.13624998703358259119022378553, 3.15304839180066403179923311610, 3.19739999332537623057432062075, 3.56542110206234700093253600260, 3.67258046559083068700654905130, 3.97406507175118712634486873192, 4.31017606298920287966247798879, 4.35778378484387692664984459778, 4.38940292218467137034842248072, 4.49941256492330002953770711382, 4.57872407996006423224708051654, 4.64252334459921917164349033753, 4.84674517702230403650068764380

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

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