Properties

Label 2-4680-1.1-c1-0-51
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 + 4·7-s − 4·11-s + 13-s − 6·17-s + 4·23-s + 25-s + 6·29-s − 8·31-s − 4·35-s − 2·37-s − 10·41-s − 4·43-s − 8·47-s + 9·49-s + 2·53-s + 4·55-s − 4·59-s + 14·61-s − 65-s − 12·67-s + 8·71-s − 10·73-s − 16·77-s + 4·83-s + 6·85-s − 10·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s − 1.20·11-s + 0.277·13-s − 1.45·17-s + 0.834·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.676·35-s − 0.328·37-s − 1.56·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.274·53-s + 0.539·55-s − 0.520·59-s + 1.79·61-s − 0.124·65-s − 1.46·67-s + 0.949·71-s − 1.17·73-s − 1.82·77-s + 0.439·83-s + 0.650·85-s − 1.05·89-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: $\chi_{4680} (1, \cdot )$
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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−8.180604414497567589384264482821, −7.23465965688715827868346322721, −6.70429518063638549972823831992, −5.48630724706287589415453698556, −4.94348202561904907438355497303, −4.39520349459478338579537402434, −3.32240915229117906302579134756, −2.33364735907848921793197115976, −1.47206787733481655449551975299, 0, 1.47206787733481655449551975299, 2.33364735907848921793197115976, 3.32240915229117906302579134756, 4.39520349459478338579537402434, 4.94348202561904907438355497303, 5.48630724706287589415453698556, 6.70429518063638549972823831992, 7.23465965688715827868346322721, 8.180604414497567589384264482821

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