Properties

Label 2-6012-1.1-c1-0-61
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  + 3.17·5-s − 0.108·7-s − 1.62·11-s − 5.77·13-s + 4.70·17-s − 5.48·19-s − 0.0951·23-s + 5.07·25-s − 4.83·29-s − 8.21·31-s − 0.343·35-s + 5.57·37-s + 3.02·41-s + 6.54·43-s − 3.97·47-s − 6.98·49-s + 5.77·53-s − 5.16·55-s + 3.26·59-s − 2.91·61-s − 18.3·65-s − 11.7·67-s − 7.35·71-s − 4.72·73-s + 0.176·77-s − 7.99·79-s − 9.73·83-s + ⋯
L(s)  = 1  + 1.41·5-s − 0.0408·7-s − 0.491·11-s − 1.60·13-s + 1.14·17-s − 1.25·19-s − 0.0198·23-s + 1.01·25-s − 0.898·29-s − 1.47·31-s − 0.0580·35-s + 0.917·37-s + 0.472·41-s + 0.998·43-s − 0.580·47-s − 0.998·49-s + 0.793·53-s − 0.696·55-s + 0.425·59-s − 0.372·61-s − 2.27·65-s − 1.44·67-s − 0.872·71-s − 0.553·73-s + 0.0200·77-s − 0.899·79-s − 1.06·83-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 \( 1 - 3.17T + 5T^{2} \)
7 \( 1 + 0.108T + 7T^{2} \)
11 \( 1 + 1.62T + 11T^{2} \)
13 \( 1 + 5.77T + 13T^{2} \)
17 \( 1 - 4.70T + 17T^{2} \)
19 \( 1 + 5.48T + 19T^{2} \)
23 \( 1 + 0.0951T + 23T^{2} \)
29 \( 1 + 4.83T + 29T^{2} \)
31 \( 1 + 8.21T + 31T^{2} \)
37 \( 1 - 5.57T + 37T^{2} \)
41 \( 1 - 3.02T + 41T^{2} \)
43 \( 1 - 6.54T + 43T^{2} \)
47 \( 1 + 3.97T + 47T^{2} \)
53 \( 1 - 5.77T + 53T^{2} \)
59 \( 1 - 3.26T + 59T^{2} \)
61 \( 1 + 2.91T + 61T^{2} \)
67 \( 1 + 11.7T + 67T^{2} \)
71 \( 1 + 7.35T + 71T^{2} \)
73 \( 1 + 4.72T + 73T^{2} \)
79 \( 1 + 7.99T + 79T^{2} \)
83 \( 1 + 9.73T + 83T^{2} \)
89 \( 1 + 9.42T + 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−7.53992496913607452067992764587, −7.15961390642464964790476754968, −6.04925265320697299646586434961, −5.69789795337286320525559447482, −4.98918801695577265625307503768, −4.15943955433932274589675613831, −2.95254979856778706298408541367, −2.30557341048299568776556791392, −1.55365766687917556057077543065, 0, 1.55365766687917556057077543065, 2.30557341048299568776556791392, 2.95254979856778706298408541367, 4.15943955433932274589675613831, 4.98918801695577265625307503768, 5.69789795337286320525559447482, 6.04925265320697299646586434961, 7.15961390642464964790476754968, 7.53992496913607452067992764587

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