Properties

Label 2-6024-1.1-c1-0-113
Degree $2$
Conductor $6024$
Sign $-1$
Analytic cond. $48.1018$
Root an. cond. $6.93555$
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-s + 0.624·5-s + 0.240·7-s + 9-s − 5.05·11-s + 4.23·13-s + 0.624·15-s + 1.70·17-s − 5.35·19-s + 0.240·21-s + 3.26·23-s − 4.60·25-s + 27-s − 8.98·29-s + 2.00·31-s − 5.05·33-s + 0.150·35-s − 5.86·37-s + 4.23·39-s − 10.7·41-s − 2.48·43-s + 0.624·45-s + 2.29·47-s − 6.94·49-s + 1.70·51-s + 10.5·53-s − 3.15·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.279·5-s + 0.0908·7-s + 0.333·9-s − 1.52·11-s + 1.17·13-s + 0.161·15-s + 0.413·17-s − 1.22·19-s + 0.0524·21-s + 0.680·23-s − 0.921·25-s + 0.192·27-s − 1.66·29-s + 0.360·31-s − 0.879·33-s + 0.0253·35-s − 0.964·37-s + 0.677·39-s − 1.68·41-s − 0.378·43-s + 0.0931·45-s + 0.334·47-s − 0.991·49-s + 0.238·51-s + 1.45·53-s − 0.425·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6024\)    =    \(2^{3} \cdot 3 \cdot 251\)
Sign: $-1$
Analytic conductor: \(48.1018\)
Root analytic conductor: \(6.93555\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6024,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
251 \( 1 - T \)
good5 \( 1 - 0.624T + 5T^{2} \)
7 \( 1 - 0.240T + 7T^{2} \)
11 \( 1 + 5.05T + 11T^{2} \)
13 \( 1 - 4.23T + 13T^{2} \)
17 \( 1 - 1.70T + 17T^{2} \)
19 \( 1 + 5.35T + 19T^{2} \)
23 \( 1 - 3.26T + 23T^{2} \)
29 \( 1 + 8.98T + 29T^{2} \)
31 \( 1 - 2.00T + 31T^{2} \)
37 \( 1 + 5.86T + 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 + 2.48T + 43T^{2} \)
47 \( 1 - 2.29T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + 1.19T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 - 8.18T + 71T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 - 7.05T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 + 9.07T + 89T^{2} \)
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

−7.86567997862821912942371879450, −7.12595225876339536339161782917, −6.28206029051796513541740594581, −5.54810867194274606566079507617, −4.90760493692018921773293001210, −3.86718238913730281453052872063, −3.25609829871814992049119108372, −2.28029655521994359855697968402, −1.56499410562770744351001599484, 0, 1.56499410562770744351001599484, 2.28029655521994359855697968402, 3.25609829871814992049119108372, 3.86718238913730281453052872063, 4.90760493692018921773293001210, 5.54810867194274606566079507617, 6.28206029051796513541740594581, 7.12595225876339536339161782917, 7.86567997862821912942371879450

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