Properties

Label 2-6024-1.1-c1-0-109
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 + 1.93·5-s + 2.74·7-s + 9-s − 0.761·11-s − 0.823·13-s − 1.93·15-s − 3.77·17-s + 2.01·19-s − 2.74·21-s − 7.45·23-s − 1.24·25-s − 27-s − 5.07·29-s + 10.0·31-s + 0.761·33-s + 5.31·35-s + 9.29·37-s + 0.823·39-s − 9.81·41-s − 11.3·43-s + 1.93·45-s − 12.1·47-s + 0.535·49-s + 3.77·51-s + 0.223·53-s − 1.47·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.866·5-s + 1.03·7-s + 0.333·9-s − 0.229·11-s − 0.228·13-s − 0.500·15-s − 0.916·17-s + 0.461·19-s − 0.599·21-s − 1.55·23-s − 0.249·25-s − 0.192·27-s − 0.942·29-s + 1.80·31-s + 0.132·33-s + 0.899·35-s + 1.52·37-s + 0.131·39-s − 1.53·41-s − 1.73·43-s + 0.288·45-s − 1.77·47-s + 0.0764·49-s + 0.529·51-s + 0.0307·53-s − 0.198·55-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 \( 1 - 1.93T + 5T^{2} \)
7 \( 1 - 2.74T + 7T^{2} \)
11 \( 1 + 0.761T + 11T^{2} \)
13 \( 1 + 0.823T + 13T^{2} \)
17 \( 1 + 3.77T + 17T^{2} \)
19 \( 1 - 2.01T + 19T^{2} \)
23 \( 1 + 7.45T + 23T^{2} \)
29 \( 1 + 5.07T + 29T^{2} \)
31 \( 1 - 10.0T + 31T^{2} \)
37 \( 1 - 9.29T + 37T^{2} \)
41 \( 1 + 9.81T + 41T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 - 0.223T + 53T^{2} \)
59 \( 1 + 6.47T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 + 5.53T + 67T^{2} \)
71 \( 1 - 1.82T + 71T^{2} \)
73 \( 1 + 5.35T + 73T^{2} \)
79 \( 1 - 4.27T + 79T^{2} \)
83 \( 1 + 1.53T + 83T^{2} \)
89 \( 1 + 0.918T + 89T^{2} \)
97 \( 1 + 6.91T + 97T^{2} \)
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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

−7.989643299603307420909493044920, −6.84090224364660709745171233479, −6.23033145807018119891808582598, −5.62289892771754236637959378227, −4.79870931255452825284692796585, −4.40542484997865032579559388418, −3.14528971643114779807072377102, −2.02326782252240873751244713594, −1.53961263304277966937764343922, 0, 1.53961263304277966937764343922, 2.02326782252240873751244713594, 3.14528971643114779807072377102, 4.40542484997865032579559388418, 4.79870931255452825284692796585, 5.62289892771754236637959378227, 6.23033145807018119891808582598, 6.84090224364660709745171233479, 7.989643299603307420909493044920

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