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Properties

Label 2-3024-7.6-c0-0-0
Degree $2$
Conductor $3024$
Sign $0.866 - 0.5i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)7-s + 1.73i·13-s + 1.73i·19-s + 25-s + 37-s + 2·43-s + (−0.499 + 0.866i)49-s − 1.73i·61-s + 67-s − 1.73i·73-s + 79-s + (1.49 − 0.866i)91-s + 1.73i·97-s + 1.73i·103-s − 2·109-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + 1.73i·13-s + 1.73i·19-s + 25-s + 37-s + 2·43-s + (−0.499 + 0.866i)49-s − 1.73i·61-s + 67-s − 1.73i·73-s + 79-s + (1.49 − 0.866i)91-s + 1.73i·97-s + 1.73i·103-s − 2·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.866 - 0.5i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ 0.866 - 0.5i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.149398737\)
\(L(\frac12)\) \(\approx\) \(1.149398737\)
\(L(1)\) 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 \( 1 \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.73iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.73iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 2T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.73iT - T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 1.73iT - T^{2} \)
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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

−9.231952526589615441386440046399, −8.082735506986688417422599023198, −7.50107329758709604015784952080, −6.58487359909858102533544541702, −6.21396040736262620704437365651, −5.01556348073403716034385179220, −4.12467176341821749925004934235, −3.61901280027233308245000798249, −2.34546185061811604126401912871, −1.22741248519183644696464509104, 0.823787029585530907052410578132, 2.65625419140775277121355884862, 2.85114296014715157245899959967, 4.14866328473362401026816035542, 5.18607485000502875041327860674, 5.68566420131456498862158535197, 6.56724020099952961111445792415, 7.33765597200931987853437932114, 8.171246182708800082430442292849, 8.887670567033321824303805971883

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