Properties

Label 2-6012-1.1-c1-0-23
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.39·5-s + 4.25·7-s + 3.66·11-s − 2.28·13-s + 5.79·17-s + 4.93·19-s + 1.43·23-s − 3.06·25-s + 3.08·29-s − 7.83·31-s − 5.92·35-s + 1.91·37-s + 1.76·41-s + 0.276·43-s + 9.79·47-s + 11.1·49-s + 10.1·53-s − 5.09·55-s + 13.3·59-s − 5.62·61-s + 3.18·65-s − 9.11·67-s − 13.3·71-s + 3.42·73-s + 15.5·77-s + 5.57·79-s − 7.04·83-s + ⋯
L(s)  = 1  − 0.622·5-s + 1.60·7-s + 1.10·11-s − 0.634·13-s + 1.40·17-s + 1.13·19-s + 0.300·23-s − 0.612·25-s + 0.572·29-s − 1.40·31-s − 1.00·35-s + 0.314·37-s + 0.275·41-s + 0.0421·43-s + 1.42·47-s + 1.58·49-s + 1.39·53-s − 0.686·55-s + 1.73·59-s − 0.719·61-s + 0.394·65-s − 1.11·67-s − 1.58·71-s + 0.401·73-s + 1.77·77-s + 0.627·79-s − 0.773·83-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: \(0\)
Selberg data: \((2,\ 6012,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.573453468\)
\(L(\frac12)\) \(\approx\) \(2.573453468\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.39T + 5T^{2} \)
7 \( 1 - 4.25T + 7T^{2} \)
11 \( 1 - 3.66T + 11T^{2} \)
13 \( 1 + 2.28T + 13T^{2} \)
17 \( 1 - 5.79T + 17T^{2} \)
19 \( 1 - 4.93T + 19T^{2} \)
23 \( 1 - 1.43T + 23T^{2} \)
29 \( 1 - 3.08T + 29T^{2} \)
31 \( 1 + 7.83T + 31T^{2} \)
37 \( 1 - 1.91T + 37T^{2} \)
41 \( 1 - 1.76T + 41T^{2} \)
43 \( 1 - 0.276T + 43T^{2} \)
47 \( 1 - 9.79T + 47T^{2} \)
53 \( 1 - 10.1T + 53T^{2} \)
59 \( 1 - 13.3T + 59T^{2} \)
61 \( 1 + 5.62T + 61T^{2} \)
67 \( 1 + 9.11T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 - 3.42T + 73T^{2} \)
79 \( 1 - 5.57T + 79T^{2} \)
83 \( 1 + 7.04T + 83T^{2} \)
89 \( 1 + 3.10T + 89T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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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

−7.86721322359554443508434711640, −7.54151979060927856002258237214, −6.93714966578250704465546329906, −5.62230995296548501648976592084, −5.34547720660093261112291884168, −4.31353688878121553234707107717, −3.85184024635726378942180995246, −2.80736936092075934964945395079, −1.65490934222848407330032856447, −0.930422722283315848559123778629, 0.930422722283315848559123778629, 1.65490934222848407330032856447, 2.80736936092075934964945395079, 3.85184024635726378942180995246, 4.31353688878121553234707107717, 5.34547720660093261112291884168, 5.62230995296548501648976592084, 6.93714966578250704465546329906, 7.54151979060927856002258237214, 7.86721322359554443508434711640

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