Properties

Label 2-6012-1.1-c1-0-63
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  − 0.399·5-s + 2.09·7-s + 4.31·11-s + 4.27·13-s − 5.86·17-s − 0.918·19-s − 7.26·23-s − 4.84·25-s − 8.94·29-s − 3.68·31-s − 0.838·35-s − 8.80·37-s + 1.00·41-s + 7.32·43-s − 3.15·47-s − 2.60·49-s − 3.31·53-s − 1.72·55-s − 0.721·59-s − 12.2·61-s − 1.71·65-s + 11.0·67-s − 13.2·71-s − 12.1·73-s + 9.04·77-s + 0.0176·79-s − 2.24·83-s + ⋯
L(s)  = 1  − 0.178·5-s + 0.792·7-s + 1.30·11-s + 1.18·13-s − 1.42·17-s − 0.210·19-s − 1.51·23-s − 0.968·25-s − 1.66·29-s − 0.661·31-s − 0.141·35-s − 1.44·37-s + 0.156·41-s + 1.11·43-s − 0.460·47-s − 0.371·49-s − 0.455·53-s − 0.232·55-s − 0.0939·59-s − 1.56·61-s − 0.212·65-s + 1.35·67-s − 1.57·71-s − 1.42·73-s + 1.03·77-s + 0.00198·79-s − 0.246·83-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 \( 1 + 0.399T + 5T^{2} \)
7 \( 1 - 2.09T + 7T^{2} \)
11 \( 1 - 4.31T + 11T^{2} \)
13 \( 1 - 4.27T + 13T^{2} \)
17 \( 1 + 5.86T + 17T^{2} \)
19 \( 1 + 0.918T + 19T^{2} \)
23 \( 1 + 7.26T + 23T^{2} \)
29 \( 1 + 8.94T + 29T^{2} \)
31 \( 1 + 3.68T + 31T^{2} \)
37 \( 1 + 8.80T + 37T^{2} \)
41 \( 1 - 1.00T + 41T^{2} \)
43 \( 1 - 7.32T + 43T^{2} \)
47 \( 1 + 3.15T + 47T^{2} \)
53 \( 1 + 3.31T + 53T^{2} \)
59 \( 1 + 0.721T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 - 0.0176T + 79T^{2} \)
83 \( 1 + 2.24T + 83T^{2} \)
89 \( 1 + 5.28T + 89T^{2} \)
97 \( 1 - 7.03T + 97T^{2} \)
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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

−7.72608704728368039764931675350, −7.07521002524484745765974839171, −6.14648536395805846561695243842, −5.83149898538363417700877979069, −4.62622584064028324501224640173, −4.02858814523544706465452220034, −3.50678720878590491752421427313, −1.96621258993475261290530329663, −1.59665424577840920866175202719, 0, 1.59665424577840920866175202719, 1.96621258993475261290530329663, 3.50678720878590491752421427313, 4.02858814523544706465452220034, 4.62622584064028324501224640173, 5.83149898538363417700877979069, 6.14648536395805846561695243842, 7.07521002524484745765974839171, 7.72608704728368039764931675350

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