Properties

Label 2-8020-1.1-c1-0-17
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  − 2.73·3-s − 5-s − 3.46·7-s + 4.46·9-s + 3.46·11-s − 6.73·13-s + 2.73·15-s + 2.73·17-s + 6.92·19-s + 9.46·21-s + 8.19·23-s + 25-s − 3.99·27-s + 5.46·29-s − 5.46·31-s − 9.46·33-s + 3.46·35-s − 0.196·37-s + 18.3·39-s − 12.3·41-s − 2·43-s − 4.46·45-s − 8.92·47-s + 4.99·49-s − 7.46·51-s + 12.1·53-s − 3.46·55-s + ⋯
L(s)  = 1  − 1.57·3-s − 0.447·5-s − 1.30·7-s + 1.48·9-s + 1.04·11-s − 1.86·13-s + 0.705·15-s + 0.662·17-s + 1.58·19-s + 2.06·21-s + 1.70·23-s + 0.200·25-s − 0.769·27-s + 1.01·29-s − 0.981·31-s − 1.64·33-s + 0.585·35-s − 0.0322·37-s + 2.94·39-s − 1.93·41-s − 0.304·43-s − 0.665·45-s − 1.30·47-s + 0.714·49-s − 1.04·51-s + 1.67·53-s − 0.467·55-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

\(L(1)\) \(\approx\) \(0.5881669100\)
\(L(\frac12)\) \(\approx\) \(0.5881669100\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.73T + 3T^{2} \)
7 \( 1 + 3.46T + 7T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 + 6.73T + 13T^{2} \)
17 \( 1 - 2.73T + 17T^{2} \)
19 \( 1 - 6.92T + 19T^{2} \)
23 \( 1 - 8.19T + 23T^{2} \)
29 \( 1 - 5.46T + 29T^{2} \)
31 \( 1 + 5.46T + 31T^{2} \)
37 \( 1 + 0.196T + 37T^{2} \)
41 \( 1 + 12.3T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 8.92T + 47T^{2} \)
53 \( 1 - 12.1T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 8.92T + 61T^{2} \)
67 \( 1 + 8.19T + 67T^{2} \)
71 \( 1 - 2.53T + 71T^{2} \)
73 \( 1 + 0.928T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−7.29684120061152659578309552910, −7.06325261747275529071148803104, −6.59851075117276055085247849550, −5.64971065936200033759038193575, −5.14133464191926448945115820768, −4.55854777703623690564683632452, −3.44482908635180680346859506415, −2.94462329159905247246810262698, −1.35341986370336354540169973606, −0.44951363442382159503491089694, 0.44951363442382159503491089694, 1.35341986370336354540169973606, 2.94462329159905247246810262698, 3.44482908635180680346859506415, 4.55854777703623690564683632452, 5.14133464191926448945115820768, 5.64971065936200033759038193575, 6.59851075117276055085247849550, 7.06325261747275529071148803104, 7.29684120061152659578309552910

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