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Properties

Label 16-700e8-1.1-c0e8-0-1
Degree $16$
Conductor $5.765\times 10^{22}$
Sign $1$
Analytic cond. $0.000221840$
Root an. cond. $0.591054$
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  + 16-s + 12·61-s − 81-s − 12·101-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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  + 16-s + 12·61-s − 81-s − 12·101-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(0.000221840\)
Root analytic conductor: \(0.591054\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5451702078\)
\(L(\frac12)\) \(\approx\) \(0.5451702078\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} \)
5 \( 1 \)
7 \( 1 - T^{4} + T^{8} \)
good3 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
11 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
13 \( ( 1 + T^{4} )^{4} \)
17 \( ( 1 - T^{4} + T^{8} )^{2} \)
19 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
23 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
29 \( ( 1 - T^{2} + T^{4} )^{4} \)
31 \( ( 1 - T^{2} + T^{4} )^{4} \)
37 \( ( 1 - T^{4} + T^{8} )^{2} \)
41 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
43 \( ( 1 - T^{4} + T^{8} )^{2} \)
47 \( ( 1 - T^{4} + T^{8} )^{2} \)
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 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
71 \( ( 1 - T )^{8}( 1 + T )^{8} \)
73 \( ( 1 - T^{4} + T^{8} )^{2} \)
79 \( ( 1 - T^{2} + T^{4} )^{4} \)
83 \( ( 1 - T^{4} + T^{8} )^{2} \)
89 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
97 \( ( 1 + T^{4} )^{4} \)
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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

−4.97034235313518680131673916278, −4.46665778558728879458703850106, −4.40510366695514855028385869107, −4.35560229737969448247255841469, −4.13176739420658555138731924910, −4.13066248015652618089986258404, −4.02795197713944862910319567984, −3.98174293552171557589671434373, −3.65369681191672722449734068898, −3.54966038497482328875417548074, −3.40492931718971685659854767103, −3.39575440047677030332807975179, −3.21083210508558674995845129852, −2.81786966784735972763852593500, −2.73088878189420697766663729175, −2.51864350358236475832139659836, −2.49328113025985959878365685198, −2.30808971342957337638530372338, −2.29020810053866576943474903554, −2.06628930928276043501562760265, −1.69905516496417531788434697241, −1.41964317179010972080787827297, −1.16108403800718222692595285252, −1.06083887717798440944089107709, −1.02094521967963520497415484235, 1.02094521967963520497415484235, 1.06083887717798440944089107709, 1.16108403800718222692595285252, 1.41964317179010972080787827297, 1.69905516496417531788434697241, 2.06628930928276043501562760265, 2.29020810053866576943474903554, 2.30808971342957337638530372338, 2.49328113025985959878365685198, 2.51864350358236475832139659836, 2.73088878189420697766663729175, 2.81786966784735972763852593500, 3.21083210508558674995845129852, 3.39575440047677030332807975179, 3.40492931718971685659854767103, 3.54966038497482328875417548074, 3.65369681191672722449734068898, 3.98174293552171557589671434373, 4.02795197713944862910319567984, 4.13066248015652618089986258404, 4.13176739420658555138731924910, 4.35560229737969448247255841469, 4.40510366695514855028385869107, 4.46665778558728879458703850106, 4.97034235313518680131673916278

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

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