Properties

Label 2-8280-1.1-c1-0-81
Degree $2$
Conductor $8280$
Sign $-1$
Analytic cond. $66.1161$
Root an. cond. $8.13118$
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  − 5-s + 2·7-s − 4·11-s + 2·13-s + 2·17-s − 8·19-s − 23-s + 25-s + 4·29-s + 8·31-s − 2·35-s + 8·37-s + 4·41-s − 10·43-s − 8·47-s − 3·49-s − 6·53-s + 4·55-s − 10·59-s + 10·61-s − 2·65-s + 2·67-s − 2·71-s + 2·73-s − 8·77-s + 16·79-s + 12·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.755·7-s − 1.20·11-s + 0.554·13-s + 0.485·17-s − 1.83·19-s − 0.208·23-s + 1/5·25-s + 0.742·29-s + 1.43·31-s − 0.338·35-s + 1.31·37-s + 0.624·41-s − 1.52·43-s − 1.16·47-s − 3/7·49-s − 0.824·53-s + 0.539·55-s − 1.30·59-s + 1.28·61-s − 0.248·65-s + 0.244·67-s − 0.237·71-s + 0.234·73-s − 0.911·77-s + 1.80·79-s + 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8280\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-1$
Analytic conductor: \(66.1161\)
Root analytic conductor: \(8.13118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
23 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−7.76986196859910930189726353998, −6.61379405843925092220884012653, −6.25337390344217045262238524439, −5.19701860404870808207037752942, −4.68569198529467664866740337253, −4.01058367021528819304449214602, −3.01049763302551176117126573989, −2.28335328675673742864312113123, −1.23911131035806829740524494216, 0, 1.23911131035806829740524494216, 2.28335328675673742864312113123, 3.01049763302551176117126573989, 4.01058367021528819304449214602, 4.68569198529467664866740337253, 5.19701860404870808207037752942, 6.25337390344217045262238524439, 6.61379405843925092220884012653, 7.76986196859910930189726353998

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