Properties

Label 2-6024-1.1-c1-0-100
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 + 3.91·5-s − 2.30·7-s + 9-s − 0.569·11-s + 0.0160·13-s − 3.91·15-s + 1.39·17-s + 0.737·19-s + 2.30·21-s − 8.19·23-s + 10.3·25-s − 27-s − 3.58·29-s − 7.46·31-s + 0.569·33-s − 9.03·35-s + 3.43·37-s − 0.0160·39-s + 4.10·41-s − 11.0·43-s + 3.91·45-s + 1.03·47-s − 1.68·49-s − 1.39·51-s + 3.38·53-s − 2.23·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.75·5-s − 0.871·7-s + 0.333·9-s − 0.171·11-s + 0.00446·13-s − 1.01·15-s + 0.338·17-s + 0.169·19-s + 0.503·21-s − 1.70·23-s + 2.07·25-s − 0.192·27-s − 0.664·29-s − 1.34·31-s + 0.0991·33-s − 1.52·35-s + 0.564·37-s − 0.00257·39-s + 0.641·41-s − 1.68·43-s + 0.584·45-s + 0.150·47-s − 0.240·49-s − 0.195·51-s + 0.465·53-s − 0.301·55-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 \( 1 - 3.91T + 5T^{2} \)
7 \( 1 + 2.30T + 7T^{2} \)
11 \( 1 + 0.569T + 11T^{2} \)
13 \( 1 - 0.0160T + 13T^{2} \)
17 \( 1 - 1.39T + 17T^{2} \)
19 \( 1 - 0.737T + 19T^{2} \)
23 \( 1 + 8.19T + 23T^{2} \)
29 \( 1 + 3.58T + 29T^{2} \)
31 \( 1 + 7.46T + 31T^{2} \)
37 \( 1 - 3.43T + 37T^{2} \)
41 \( 1 - 4.10T + 41T^{2} \)
43 \( 1 + 11.0T + 43T^{2} \)
47 \( 1 - 1.03T + 47T^{2} \)
53 \( 1 - 3.38T + 53T^{2} \)
59 \( 1 - 0.600T + 59T^{2} \)
61 \( 1 + 13.9T + 61T^{2} \)
67 \( 1 - 9.60T + 67T^{2} \)
71 \( 1 - 4.10T + 71T^{2} \)
73 \( 1 + 13.6T + 73T^{2} \)
79 \( 1 - 8.57T + 79T^{2} \)
83 \( 1 + 1.99T + 83T^{2} \)
89 \( 1 + 18.0T + 89T^{2} \)
97 \( 1 + 8.09T + 97T^{2} \)
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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

−7.57849309564505421959438448137, −6.77873087655787145804117289891, −6.14094474612910375870991628790, −5.72510466848509073627439497432, −5.14815610645276218316854471577, −4.07191544910219738473770012632, −3.12924681048459210415840498374, −2.17767330576502184595008694377, −1.45501288369930059287641551629, 0, 1.45501288369930059287641551629, 2.17767330576502184595008694377, 3.12924681048459210415840498374, 4.07191544910219738473770012632, 5.14815610645276218316854471577, 5.72510466848509073627439497432, 6.14094474612910375870991628790, 6.77873087655787145804117289891, 7.57849309564505421959438448137

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