Properties

Label 2-4680-1.1-c1-0-57
Degree $2$
Conductor $4680$
Sign $-1$
Analytic cond. $37.3699$
Root an. cond. $6.11309$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 3·7-s − 5·11-s − 13-s + 3·17-s − 8·19-s + 3·23-s + 25-s + 2·29-s − 10·31-s + 3·35-s − 3·37-s − 3·41-s − 10·43-s + 6·47-s + 2·49-s − 5·53-s − 5·55-s − 4·59-s − 7·61-s − 65-s + 12·67-s − 5·71-s + 2·73-s − 15·77-s − 13·79-s + 12·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.13·7-s − 1.50·11-s − 0.277·13-s + 0.727·17-s − 1.83·19-s + 0.625·23-s + 1/5·25-s + 0.371·29-s − 1.79·31-s + 0.507·35-s − 0.493·37-s − 0.468·41-s − 1.52·43-s + 0.875·47-s + 2/7·49-s − 0.686·53-s − 0.674·55-s − 0.520·59-s − 0.896·61-s − 0.124·65-s + 1.46·67-s − 0.593·71-s + 0.234·73-s − 1.70·77-s − 1.46·79-s + 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4680\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(37.3699\)
Root analytic conductor: \(6.11309\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4680,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
13 \( 1 + T \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−8.040300817419889487340992272551, −7.32350601773238389640751272254, −6.51636016703422468528885784305, −5.48927472426394427749267003240, −5.13124238045700232058062769052, −4.36431631260615805790249240018, −3.22898632671117123034544099556, −2.27512072123423792723142770388, −1.59234257892317960562059000064, 0, 1.59234257892317960562059000064, 2.27512072123423792723142770388, 3.22898632671117123034544099556, 4.36431631260615805790249240018, 5.13124238045700232058062769052, 5.48927472426394427749267003240, 6.51636016703422468528885784305, 7.32350601773238389640751272254, 8.040300817419889487340992272551

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