Properties

Degree $2$
Conductor $6012$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 4.48·7-s + 1.62·11-s + 3.38·13-s + 5.30·17-s + 6.97·19-s − 0.132·23-s − 25-s + 7.86·29-s + 10.5·31-s − 8.97·35-s + 1.72·37-s − 8.40·41-s − 6.76·43-s − 0.208·47-s + 13.1·49-s − 8.68·53-s − 3.25·55-s − 13.8·59-s − 3.06·61-s − 6.76·65-s + 7.51·67-s − 2.06·71-s + 5.43·73-s + 7.28·77-s − 17.5·79-s + 12.1·83-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.69·7-s + 0.489·11-s + 0.938·13-s + 1.28·17-s + 1.59·19-s − 0.0276·23-s − 0.200·25-s + 1.46·29-s + 1.90·31-s − 1.51·35-s + 0.282·37-s − 1.31·41-s − 1.03·43-s − 0.0304·47-s + 1.87·49-s − 1.19·53-s − 0.438·55-s − 1.79·59-s − 0.392·61-s − 0.839·65-s + 0.917·67-s − 0.245·71-s + 0.636·73-s + 0.830·77-s − 1.97·79-s + 1.33·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6012\)    =    \(2^{2} \cdot 3^{2} \cdot 167\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{6012} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6012,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.699489720\)
\(L(\frac12)\) \(\approx\) \(2.699489720\)
\(L(\frac{3}{2})\) 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 \( 1 \)
167 \( 1 - T \)
good5 \( 1 + 2T + 5T^{2} \)
7 \( 1 - 4.48T + 7T^{2} \)
11 \( 1 - 1.62T + 11T^{2} \)
13 \( 1 - 3.38T + 13T^{2} \)
17 \( 1 - 5.30T + 17T^{2} \)
19 \( 1 - 6.97T + 19T^{2} \)
23 \( 1 + 0.132T + 23T^{2} \)
29 \( 1 - 7.86T + 29T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 - 1.72T + 37T^{2} \)
41 \( 1 + 8.40T + 41T^{2} \)
43 \( 1 + 6.76T + 43T^{2} \)
47 \( 1 + 0.208T + 47T^{2} \)
53 \( 1 + 8.68T + 53T^{2} \)
59 \( 1 + 13.8T + 59T^{2} \)
61 \( 1 + 3.06T + 61T^{2} \)
67 \( 1 - 7.51T + 67T^{2} \)
71 \( 1 + 2.06T + 71T^{2} \)
73 \( 1 - 5.43T + 73T^{2} \)
79 \( 1 + 17.5T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + 4.21T + 97T^{2} \)
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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

−8.139370321916547448356338252082, −7.64916710369540616798296526547, −6.74073660806361604274061799525, −5.90338114465707527155486028019, −4.99018735138118508767460016404, −4.58336815585772546741611115094, −3.62468217508920656778470352265, −2.98856155712789124660182847044, −1.50660874484110468289013334227, −1.01584197122131575960536029100, 1.01584197122131575960536029100, 1.50660874484110468289013334227, 2.98856155712789124660182847044, 3.62468217508920656778470352265, 4.58336815585772546741611115094, 4.99018735138118508767460016404, 5.90338114465707527155486028019, 6.74073660806361604274061799525, 7.64916710369540616798296526547, 8.139370321916547448356338252082

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