Properties

Label 2-6012-1.1-c1-0-52
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  − 1.90·5-s + 2.81·7-s − 0.318·11-s + 5.12·13-s − 3.73·17-s − 0.725·19-s − 0.612·23-s − 1.36·25-s + 3.87·29-s − 6.65·31-s − 5.36·35-s − 9.04·37-s − 9.55·41-s − 10.9·43-s − 8.76·47-s + 0.915·49-s + 2.46·53-s + 0.608·55-s + 11.0·59-s + 12.5·61-s − 9.77·65-s + 5.21·67-s + 11.3·71-s + 2.88·73-s − 0.897·77-s + 8.98·79-s − 4.04·83-s + ⋯
L(s)  = 1  − 0.853·5-s + 1.06·7-s − 0.0961·11-s + 1.42·13-s − 0.905·17-s − 0.166·19-s − 0.127·23-s − 0.272·25-s + 0.720·29-s − 1.19·31-s − 0.907·35-s − 1.48·37-s − 1.49·41-s − 1.66·43-s − 1.27·47-s + 0.130·49-s + 0.338·53-s + 0.0820·55-s + 1.44·59-s + 1.60·61-s − 1.21·65-s + 0.637·67-s + 1.34·71-s + 0.337·73-s − 0.102·77-s + 1.01·79-s − 0.444·83-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 \( 1 + 1.90T + 5T^{2} \)
7 \( 1 - 2.81T + 7T^{2} \)
11 \( 1 + 0.318T + 11T^{2} \)
13 \( 1 - 5.12T + 13T^{2} \)
17 \( 1 + 3.73T + 17T^{2} \)
19 \( 1 + 0.725T + 19T^{2} \)
23 \( 1 + 0.612T + 23T^{2} \)
29 \( 1 - 3.87T + 29T^{2} \)
31 \( 1 + 6.65T + 31T^{2} \)
37 \( 1 + 9.04T + 37T^{2} \)
41 \( 1 + 9.55T + 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 + 8.76T + 47T^{2} \)
53 \( 1 - 2.46T + 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 - 5.21T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 2.88T + 73T^{2} \)
79 \( 1 - 8.98T + 79T^{2} \)
83 \( 1 + 4.04T + 83T^{2} \)
89 \( 1 - 9.15T + 89T^{2} \)
97 \( 1 + 9.62T + 97T^{2} \)
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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

−8.005761532579491878524628664897, −6.89398598100351642635616399492, −6.56392020087178595256828961066, −5.33280076070978531478341962632, −4.93256220581892337453414578065, −3.82213246399134204474000037897, −3.59593049187975992780562017323, −2.16427006480293518396605144915, −1.38592505325482015043604847330, 0, 1.38592505325482015043604847330, 2.16427006480293518396605144915, 3.59593049187975992780562017323, 3.82213246399134204474000037897, 4.93256220581892337453414578065, 5.33280076070978531478341962632, 6.56392020087178595256828961066, 6.89398598100351642635616399492, 8.005761532579491878524628664897

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