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 − 0.516·7-s − 0.538·11-s − 0.295·13-s + 6.99·17-s − 4.06·19-s − 0.781·23-s − 25-s − 0.811·29-s + 3.07·31-s + 1.03·35-s − 3.95·37-s − 0.743·41-s + 0.590·43-s + 3.47·47-s − 6.73·49-s − 6.69·53-s + 1.07·55-s + 8.91·59-s + 4.63·61-s + 0.590·65-s + 6.55·67-s − 12.4·71-s + 7.77·73-s + 0.278·77-s − 4.88·79-s + 1.11·83-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.195·7-s − 0.162·11-s − 0.0819·13-s + 1.69·17-s − 0.932·19-s − 0.162·23-s − 0.200·25-s − 0.150·29-s + 0.552·31-s + 0.174·35-s − 0.650·37-s − 0.116·41-s + 0.0900·43-s + 0.506·47-s − 0.961·49-s − 0.920·53-s + 0.145·55-s + 1.16·59-s + 0.594·61-s + 0.0732·65-s + 0.800·67-s − 1.47·71-s + 0.910·73-s + 0.0316·77-s − 0.549·79-s + 0.122·83-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

\(L(1)\) \(\approx\) \(1.267664167\)
\(L(\frac12)\) \(\approx\) \(1.267664167\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.516T + 7T^{2} \)
11 \( 1 + 0.538T + 11T^{2} \)
13 \( 1 + 0.295T + 13T^{2} \)
17 \( 1 - 6.99T + 17T^{2} \)
19 \( 1 + 4.06T + 19T^{2} \)
23 \( 1 + 0.781T + 23T^{2} \)
29 \( 1 + 0.811T + 29T^{2} \)
31 \( 1 - 3.07T + 31T^{2} \)
37 \( 1 + 3.95T + 37T^{2} \)
41 \( 1 + 0.743T + 41T^{2} \)
43 \( 1 - 0.590T + 43T^{2} \)
47 \( 1 - 3.47T + 47T^{2} \)
53 \( 1 + 6.69T + 53T^{2} \)
59 \( 1 - 8.91T + 59T^{2} \)
61 \( 1 - 4.63T + 61T^{2} \)
67 \( 1 - 6.55T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 7.77T + 73T^{2} \)
79 \( 1 + 4.88T + 79T^{2} \)
83 \( 1 - 1.11T + 83T^{2} \)
89 \( 1 - 1.79T + 89T^{2} \)
97 \( 1 + 17.1T + 97T^{2} \)
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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

−8.070537639683897164836786744980, −7.47946161093326635449360600193, −6.74748013498605413466388370515, −5.93014195926811040038089278188, −5.21371941611930490532709229290, −4.33653068101832150671045387553, −3.64165772659627140897306247006, −2.96358500094648538473670534937, −1.82388446491274241274680364875, −0.58624321435360987084695345712, 0.58624321435360987084695345712, 1.82388446491274241274680364875, 2.96358500094648538473670534937, 3.64165772659627140897306247006, 4.33653068101832150671045387553, 5.21371941611930490532709229290, 5.93014195926811040038089278188, 6.74748013498605413466388370515, 7.47946161093326635449360600193, 8.070537639683897164836786744980

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