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Properties

Label 16-3675e8-1.1-c0e8-0-4
Degree $16$
Conductor $3.327\times 10^{28}$
Sign $1$
Analytic cond. $128.031$
Root an. cond. $1.35427$
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  − 4·2-s + 10·4-s − 24·8-s + 51·16-s − 4·23-s − 96·32-s + 16·46-s + 174·64-s − 4·79-s + 81-s − 40·92-s − 4·109-s + 8·113-s − 2·121-s + 127-s − 300·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 16·158-s − 4·162-s + 163-s + 167-s + 173-s + ⋯
L(s)  = 1  − 4·2-s + 10·4-s − 24·8-s + 51·16-s − 4·23-s − 96·32-s + 16·46-s + 174·64-s − 4·79-s + 81-s − 40·92-s − 4·109-s + 8·113-s − 2·121-s + 127-s − 300·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 16·158-s − 4·162-s + 163-s + 167-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 5^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(128.031\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02767670441\)
\(L(\frac12)\) \(\approx\) \(0.02767670441\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−3.56835048788202005722320990389, −3.41634559299908849608666396094, −3.41363568225025485266449309963, −3.38928022799992317417108140830, −3.33813754794851567005113545369, −2.99360081656079418115025653582, −2.83930274816370931881916512810, −2.70772812086220422238776770766, −2.68145551637028785903530516325, −2.59585889007618313845175639243, −2.55763881049289271673560503667, −2.46907269882651548086104327396, −2.26443371322284119698967552091, −2.15788989722645683237727338321, −1.93945323010125267684490845348, −1.86535922565505144281104896577, −1.86207091316215081724083812775, −1.49057065697647039642625626149, −1.42451708693054778897653361960, −1.30387791676171214020147488210, −1.25153430214294119963154025060, −0.76217796079752179463253421616, −0.61531571448270503956895222268, −0.50120586463911019264121723861, −0.16588293812962109539062318672, 0.16588293812962109539062318672, 0.50120586463911019264121723861, 0.61531571448270503956895222268, 0.76217796079752179463253421616, 1.25153430214294119963154025060, 1.30387791676171214020147488210, 1.42451708693054778897653361960, 1.49057065697647039642625626149, 1.86207091316215081724083812775, 1.86535922565505144281104896577, 1.93945323010125267684490845348, 2.15788989722645683237727338321, 2.26443371322284119698967552091, 2.46907269882651548086104327396, 2.55763881049289271673560503667, 2.59585889007618313845175639243, 2.68145551637028785903530516325, 2.70772812086220422238776770766, 2.83930274816370931881916512810, 2.99360081656079418115025653582, 3.33813754794851567005113545369, 3.38928022799992317417108140830, 3.41363568225025485266449309963, 3.41634559299908849608666396094, 3.56835048788202005722320990389

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

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