Properties

Label 2-8036-1.1-c1-0-18
Degree $2$
Conductor $8036$
Sign $1$
Analytic cond. $64.1677$
Root an. cond. $8.01047$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·5-s − 2·9-s + 3·11-s − 4·13-s − 3·15-s − 7·19-s + 6·23-s + 4·25-s − 5·27-s + 6·29-s − 10·31-s + 3·33-s + 2·37-s − 4·39-s − 41-s − 4·43-s + 6·45-s + 12·47-s − 6·53-s − 9·55-s − 7·57-s + 6·59-s − 13·61-s + 12·65-s − 4·67-s + 6·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s − 2/3·9-s + 0.904·11-s − 1.10·13-s − 0.774·15-s − 1.60·19-s + 1.25·23-s + 4/5·25-s − 0.962·27-s + 1.11·29-s − 1.79·31-s + 0.522·33-s + 0.328·37-s − 0.640·39-s − 0.156·41-s − 0.609·43-s + 0.894·45-s + 1.75·47-s − 0.824·53-s − 1.21·55-s − 0.927·57-s + 0.781·59-s − 1.66·61-s + 1.48·65-s − 0.488·67-s + 0.722·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8036\)    =    \(2^{2} \cdot 7^{2} \cdot 41\)
Sign: $1$
Analytic conductor: \(64.1677\)
Root analytic conductor: \(8.01047\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8036} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8036,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.095853518\)
\(L(\frac12)\) \(\approx\) \(1.095853518\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 + T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−7.85469270651828709299816671466, −7.23740498038105267998641950476, −6.67653099768424070226067723636, −5.76799574586480259364670501822, −4.78602534604741827631769270773, −4.22570864781730249376473352849, −3.49608096350530837031433405064, −2.81867535621895764273660745052, −1.90933516200630251231969048925, −0.48153827441527905807666861519, 0.48153827441527905807666861519, 1.90933516200630251231969048925, 2.81867535621895764273660745052, 3.49608096350530837031433405064, 4.22570864781730249376473352849, 4.78602534604741827631769270773, 5.76799574586480259364670501822, 6.67653099768424070226067723636, 7.23740498038105267998641950476, 7.85469270651828709299816671466

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