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Properties

Label 2-8016-1.1-c1-0-117
Degree $2$
Conductor $8016$
Sign $-1$
Analytic cond. $64.0080$
Root an. cond. $8.00050$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 2·17-s + 8·19-s + 4·23-s − 5·25-s − 27-s − 10·29-s − 8·31-s − 8·37-s + 2·41-s − 2·43-s − 7·49-s − 2·51-s − 12·53-s − 8·57-s + 6·59-s + 14·61-s + 6·67-s − 4·69-s − 12·71-s + 2·73-s + 5·75-s + 4·79-s + 81-s + 6·83-s + 10·87-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 0.485·17-s + 1.83·19-s + 0.834·23-s − 25-s − 0.192·27-s − 1.85·29-s − 1.43·31-s − 1.31·37-s + 0.312·41-s − 0.304·43-s − 49-s − 0.280·51-s − 1.64·53-s − 1.05·57-s + 0.781·59-s + 1.79·61-s + 0.733·67-s − 0.481·69-s − 1.42·71-s + 0.234·73-s + 0.577·75-s + 0.450·79-s + 1/9·81-s + 0.658·83-s + 1.07·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8016\)    =    \(2^{4} \cdot 3 \cdot 167\)
Sign: $-1$
Analytic conductor: \(64.0080\)
Root analytic conductor: \(8.00050\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
167 \( 1 + T \)
good5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.42778857982227494269096924863, −6.91679966635424091124295877584, −5.94720621315246046672289244274, −5.36331528281534880802857753573, −4.95984995871059671778250510561, −3.68599181278496489585728114353, −3.39272231870526222998191485665, −2.06218525402010333827396581641, −1.22816967262578614972876459227, 0, 1.22816967262578614972876459227, 2.06218525402010333827396581641, 3.39272231870526222998191485665, 3.68599181278496489585728114353, 4.95984995871059671778250510561, 5.36331528281534880802857753573, 5.94720621315246046672289244274, 6.91679966635424091124295877584, 7.42778857982227494269096924863

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