Properties

Label 2-8020-1.1-c1-0-11
Degree $2$
Conductor $8020$
Sign $1$
Analytic cond. $64.0400$
Root an. cond. $8.00250$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.478·3-s − 5-s + 1.59·7-s − 2.77·9-s − 6.51·11-s − 3.37·13-s + 0.478·15-s − 3.23·17-s − 0.00948·19-s − 0.762·21-s + 8.46·23-s + 25-s + 2.76·27-s + 0.667·29-s − 2.17·31-s + 3.12·33-s − 1.59·35-s − 9.25·37-s + 1.61·39-s − 5.04·41-s − 5.75·43-s + 2.77·45-s + 3.87·47-s − 4.46·49-s + 1.55·51-s + 3.05·53-s + 6.51·55-s + ⋯
L(s)  = 1  − 0.276·3-s − 0.447·5-s + 0.601·7-s − 0.923·9-s − 1.96·11-s − 0.936·13-s + 0.123·15-s − 0.785·17-s − 0.00217·19-s − 0.166·21-s + 1.76·23-s + 0.200·25-s + 0.531·27-s + 0.124·29-s − 0.391·31-s + 0.543·33-s − 0.269·35-s − 1.52·37-s + 0.258·39-s − 0.788·41-s − 0.877·43-s + 0.413·45-s + 0.564·47-s − 0.638·49-s + 0.217·51-s + 0.419·53-s + 0.878·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8020\)    =    \(2^{2} \cdot 5 \cdot 401\)
Sign: $1$
Analytic conductor: \(64.0400\)
Root analytic conductor: \(8.00250\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8020,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5453752561\)
\(L(\frac12)\) \(\approx\) \(0.5453752561\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
401 \( 1 + T \)
good3 \( 1 + 0.478T + 3T^{2} \)
7 \( 1 - 1.59T + 7T^{2} \)
11 \( 1 + 6.51T + 11T^{2} \)
13 \( 1 + 3.37T + 13T^{2} \)
17 \( 1 + 3.23T + 17T^{2} \)
19 \( 1 + 0.00948T + 19T^{2} \)
23 \( 1 - 8.46T + 23T^{2} \)
29 \( 1 - 0.667T + 29T^{2} \)
31 \( 1 + 2.17T + 31T^{2} \)
37 \( 1 + 9.25T + 37T^{2} \)
41 \( 1 + 5.04T + 41T^{2} \)
43 \( 1 + 5.75T + 43T^{2} \)
47 \( 1 - 3.87T + 47T^{2} \)
53 \( 1 - 3.05T + 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 - 2.12T + 67T^{2} \)
71 \( 1 - 3.04T + 71T^{2} \)
73 \( 1 - 1.95T + 73T^{2} \)
79 \( 1 - 8.09T + 79T^{2} \)
83 \( 1 + 1.98T + 83T^{2} \)
89 \( 1 + 9.89T + 89T^{2} \)
97 \( 1 - 3.02T + 97T^{2} \)
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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

−7.77017384930784439827503441180, −7.27823957778641915129624719867, −6.53149961059532871496080310698, −5.46421245418607756283535713435, −5.03816592099572931428677400024, −4.65374662155682173342834866039, −3.26788735639239268022848703661, −2.77514983034011894650864431468, −1.87592970799625197033736683393, −0.34803621964698552120335681630, 0.34803621964698552120335681630, 1.87592970799625197033736683393, 2.77514983034011894650864431468, 3.26788735639239268022848703661, 4.65374662155682173342834866039, 5.03816592099572931428677400024, 5.46421245418607756283535713435, 6.53149961059532871496080310698, 7.27823957778641915129624719867, 7.77017384930784439827503441180

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