Properties

Label 2-6012-1.1-c1-0-42
Degree $2$
Conductor $6012$
Sign $-1$
Analytic cond. $48.0060$
Root an. cond. $6.92864$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.951·5-s − 1.28·7-s + 0.951·11-s − 2.82·13-s − 1.31·17-s + 5.94·19-s + 2.56·23-s − 4.09·25-s + 2.48·29-s + 0.984·31-s + 1.21·35-s − 7.81·37-s + 7.41·41-s + 7.65·43-s − 10.1·47-s − 5.35·49-s + 3.23·53-s − 0.904·55-s + 10.3·59-s − 11.8·61-s + 2.68·65-s + 8.14·67-s − 1.71·71-s − 0.502·73-s − 1.21·77-s − 14.6·79-s + 11.3·83-s + ⋯
L(s)  = 1  − 0.425·5-s − 0.484·7-s + 0.286·11-s − 0.784·13-s − 0.318·17-s + 1.36·19-s + 0.535·23-s − 0.819·25-s + 0.462·29-s + 0.176·31-s + 0.206·35-s − 1.28·37-s + 1.15·41-s + 1.16·43-s − 1.47·47-s − 0.765·49-s + 0.444·53-s − 0.122·55-s + 1.34·59-s − 1.52·61-s + 0.333·65-s + 0.995·67-s − 0.203·71-s − 0.0587·73-s − 0.138·77-s − 1.65·79-s + 1.24·83-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: $-1$
Analytic conductor: \(48.0060\)
Root analytic conductor: \(6.92864\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6012,\ (\ :1/2),\ -1)\)

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 \( 1 \)
167 \( 1 + T \)
good5 \( 1 + 0.951T + 5T^{2} \)
7 \( 1 + 1.28T + 7T^{2} \)
11 \( 1 - 0.951T + 11T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 + 1.31T + 17T^{2} \)
19 \( 1 - 5.94T + 19T^{2} \)
23 \( 1 - 2.56T + 23T^{2} \)
29 \( 1 - 2.48T + 29T^{2} \)
31 \( 1 - 0.984T + 31T^{2} \)
37 \( 1 + 7.81T + 37T^{2} \)
41 \( 1 - 7.41T + 41T^{2} \)
43 \( 1 - 7.65T + 43T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 - 3.23T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 - 8.14T + 67T^{2} \)
71 \( 1 + 1.71T + 71T^{2} \)
73 \( 1 + 0.502T + 73T^{2} \)
79 \( 1 + 14.6T + 79T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 + 6.15T + 89T^{2} \)
97 \( 1 - 4.75T + 97T^{2} \)
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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

−7.63355339256429353160511116505, −7.09781088912242508288642262145, −6.40031287491029302506406830312, −5.52674968432294625108468945511, −4.84867719900937316276912571721, −3.99389401966633799739567081359, −3.23885387181180400624779964981, −2.45739273861161578006893434248, −1.23292041884592317070890126957, 0, 1.23292041884592317070890126957, 2.45739273861161578006893434248, 3.23885387181180400624779964981, 3.99389401966633799739567081359, 4.84867719900937316276912571721, 5.52674968432294625108468945511, 6.40031287491029302506406830312, 7.09781088912242508288642262145, 7.63355339256429353160511116505

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