Properties

Label 2-6012-1.1-c1-0-45
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  − 3.67·5-s + 4.79·7-s − 1.91·11-s + 3.05·13-s + 2.92·17-s − 8.17·19-s − 6.97·23-s + 8.49·25-s + 3.87·29-s + 0.104·31-s − 17.6·35-s − 1.37·37-s + 10.4·41-s − 5.08·43-s − 10.8·47-s + 16.0·49-s − 7.42·53-s + 7.03·55-s − 0.535·59-s + 8.80·61-s − 11.2·65-s − 8.50·67-s + 5.00·71-s + 8.80·73-s − 9.18·77-s − 6.76·79-s − 10.3·83-s + ⋯
L(s)  = 1  − 1.64·5-s + 1.81·7-s − 0.577·11-s + 0.847·13-s + 0.710·17-s − 1.87·19-s − 1.45·23-s + 1.69·25-s + 0.720·29-s + 0.0187·31-s − 2.97·35-s − 0.226·37-s + 1.63·41-s − 0.774·43-s − 1.57·47-s + 2.28·49-s − 1.01·53-s + 0.948·55-s − 0.0696·59-s + 1.12·61-s − 1.39·65-s − 1.03·67-s + 0.594·71-s + 1.03·73-s − 1.04·77-s − 0.761·79-s − 1.13·83-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 \( 1 + 3.67T + 5T^{2} \)
7 \( 1 - 4.79T + 7T^{2} \)
11 \( 1 + 1.91T + 11T^{2} \)
13 \( 1 - 3.05T + 13T^{2} \)
17 \( 1 - 2.92T + 17T^{2} \)
19 \( 1 + 8.17T + 19T^{2} \)
23 \( 1 + 6.97T + 23T^{2} \)
29 \( 1 - 3.87T + 29T^{2} \)
31 \( 1 - 0.104T + 31T^{2} \)
37 \( 1 + 1.37T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + 5.08T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + 7.42T + 53T^{2} \)
59 \( 1 + 0.535T + 59T^{2} \)
61 \( 1 - 8.80T + 61T^{2} \)
67 \( 1 + 8.50T + 67T^{2} \)
71 \( 1 - 5.00T + 71T^{2} \)
73 \( 1 - 8.80T + 73T^{2} \)
79 \( 1 + 6.76T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−7.978593307315781983432266928738, −7.27838099887826477650939161377, −6.33616122184797873324052102264, −5.46864825332516911413558178206, −4.49771587407210907850533417890, −4.29176813107568924899521969117, −3.41342618228403518022843689483, −2.25262196055315800254741644641, −1.28322454468950760554651346196, 0, 1.28322454468950760554651346196, 2.25262196055315800254741644641, 3.41342618228403518022843689483, 4.29176813107568924899521969117, 4.49771587407210907850533417890, 5.46864825332516911413558178206, 6.33616122184797873324052102264, 7.27838099887826477650939161377, 7.978593307315781983432266928738

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