Properties

Label 2-8020-1.1-c1-0-94
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·3-s + 5-s + 2.72·7-s + 0.0129·9-s − 2.68·11-s + 3.26·13-s − 1.73·15-s − 1.12·17-s − 5.16·19-s − 4.72·21-s + 4.43·23-s + 25-s + 5.18·27-s − 4.37·29-s − 4.77·31-s + 4.65·33-s + 2.72·35-s + 5.04·37-s − 5.67·39-s − 5.99·41-s + 10.1·43-s + 0.0129·45-s + 1.88·47-s + 0.404·49-s + 1.95·51-s − 4.64·53-s − 2.68·55-s + ⋯
L(s)  = 1  − 1.00·3-s + 0.447·5-s + 1.02·7-s + 0.00431·9-s − 0.809·11-s + 0.906·13-s − 0.448·15-s − 0.272·17-s − 1.18·19-s − 1.03·21-s + 0.924·23-s + 0.200·25-s + 0.997·27-s − 0.811·29-s − 0.858·31-s + 0.811·33-s + 0.459·35-s + 0.828·37-s − 0.908·39-s − 0.936·41-s + 1.54·43-s + 0.00193·45-s + 0.274·47-s + 0.0578·49-s + 0.273·51-s − 0.638·53-s − 0.361·55-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: \(1\)
Selberg data: \((2,\ 8020,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
401 \( 1 + T \)
good3 \( 1 + 1.73T + 3T^{2} \)
7 \( 1 - 2.72T + 7T^{2} \)
11 \( 1 + 2.68T + 11T^{2} \)
13 \( 1 - 3.26T + 13T^{2} \)
17 \( 1 + 1.12T + 17T^{2} \)
19 \( 1 + 5.16T + 19T^{2} \)
23 \( 1 - 4.43T + 23T^{2} \)
29 \( 1 + 4.37T + 29T^{2} \)
31 \( 1 + 4.77T + 31T^{2} \)
37 \( 1 - 5.04T + 37T^{2} \)
41 \( 1 + 5.99T + 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 - 1.88T + 47T^{2} \)
53 \( 1 + 4.64T + 53T^{2} \)
59 \( 1 - 2.26T + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 + 5.23T + 71T^{2} \)
73 \( 1 - 9.75T + 73T^{2} \)
79 \( 1 - 7.93T + 79T^{2} \)
83 \( 1 + 1.07T + 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 - 3.63T + 97T^{2} \)
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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

−7.48929938858144174441329040319, −6.61168064010620274944418282450, −5.98635757397480948879044118416, −5.45874403206253279970187395419, −4.82907813665419519639143674418, −4.16753073054282170732221527701, −3.01472173783386502459443243916, −2.08170005760977306816754396542, −1.21566507171091955231492897042, 0, 1.21566507171091955231492897042, 2.08170005760977306816754396542, 3.01472173783386502459443243916, 4.16753073054282170732221527701, 4.82907813665419519639143674418, 5.45874403206253279970187395419, 5.98635757397480948879044118416, 6.61168064010620274944418282450, 7.48929938858144174441329040319

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