Properties

Label 2-8020-1.1-c1-0-21
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  − 1.70·3-s + 5-s − 1.03·7-s − 0.0914·9-s + 1.24·11-s − 4.47·13-s − 1.70·15-s + 2.95·17-s + 0.264·19-s + 1.77·21-s − 1.16·23-s + 25-s + 5.27·27-s − 2.37·29-s − 3.78·31-s − 2.12·33-s − 1.03·35-s + 10.8·37-s + 7.62·39-s − 3.50·41-s − 5.37·43-s − 0.0914·45-s − 7.48·47-s − 5.92·49-s − 5.04·51-s + 1.25·53-s + 1.24·55-s + ⋯
L(s)  = 1  − 0.984·3-s + 0.447·5-s − 0.392·7-s − 0.0304·9-s + 0.376·11-s − 1.24·13-s − 0.440·15-s + 0.717·17-s + 0.0607·19-s + 0.386·21-s − 0.242·23-s + 0.200·25-s + 1.01·27-s − 0.440·29-s − 0.679·31-s − 0.370·33-s − 0.175·35-s + 1.78·37-s + 1.22·39-s − 0.547·41-s − 0.819·43-s − 0.0136·45-s − 1.09·47-s − 0.845·49-s − 0.706·51-s + 0.173·53-s + 0.168·55-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

\(L(1)\) \(\approx\) \(0.9449184161\)
\(L(\frac12)\) \(\approx\) \(0.9449184161\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.70T + 3T^{2} \)
7 \( 1 + 1.03T + 7T^{2} \)
11 \( 1 - 1.24T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 - 2.95T + 17T^{2} \)
19 \( 1 - 0.264T + 19T^{2} \)
23 \( 1 + 1.16T + 23T^{2} \)
29 \( 1 + 2.37T + 29T^{2} \)
31 \( 1 + 3.78T + 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 + 3.50T + 41T^{2} \)
43 \( 1 + 5.37T + 43T^{2} \)
47 \( 1 + 7.48T + 47T^{2} \)
53 \( 1 - 1.25T + 53T^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 - 0.329T + 61T^{2} \)
67 \( 1 + 3.42T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 + 2.47T + 73T^{2} \)
79 \( 1 + 3.30T + 79T^{2} \)
83 \( 1 - 9.25T + 83T^{2} \)
89 \( 1 - 3.23T + 89T^{2} \)
97 \( 1 - 2.95T + 97T^{2} \)
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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

−7.68848293857400755674052174391, −7.00838149806762462112769955780, −6.33730086413765532324517171059, −5.77505304005798257010669146540, −5.15142266825322492047118227868, −4.53216852850164772155388721189, −3.46850323219259280015654938996, −2.67793563853237510802078682615, −1.66967182320403057176856939022, −0.50231741025325767934148112425, 0.50231741025325767934148112425, 1.66967182320403057176856939022, 2.67793563853237510802078682615, 3.46850323219259280015654938996, 4.53216852850164772155388721189, 5.15142266825322492047118227868, 5.77505304005798257010669146540, 6.33730086413765532324517171059, 7.00838149806762462112769955780, 7.68848293857400755674052174391

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