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Properties

Label 2-3024-84.59-c0-0-0
Degree $2$
Conductor $3024$
Sign $0.386 - 0.922i$
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 + (0.5 + 0.866i)19-s + (0.5 − 0.866i)25-s + (−0.5 + 0.866i)31-s + (1 + 1.73i)37-s + 1.73i·43-s + (−0.499 − 0.866i)49-s + (−1.5 + 0.866i)61-s + (1.5 + 0.866i)73-s − 1.73i·97-s + (1 + 1.73i)103-s + (0.5 − 0.866i)109-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)25-s + (−0.5 + 0.866i)31-s + (1 + 1.73i)37-s + 1.73i·43-s + (−0.499 − 0.866i)49-s + (−1.5 + 0.866i)61-s + (1.5 + 0.866i)73-s − 1.73i·97-s + (1 + 1.73i)103-s + (0.5 − 0.866i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.080790520\)
\(L(\frac12)\) \(\approx\) \(1.080790520\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.73iT - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
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

−9.033078272488603153747116717778, −8.321267328526487072880160028881, −7.66444432225361500905678299622, −6.59417055961223079887456440304, −6.11647421060675859243536855654, −5.24646355096021408915491488884, −4.43862406385607811620286990188, −3.30211812971448327960015430547, −2.65280056791359259628016203421, −1.39873304819395226299059943892, 0.71050136334443935450162667755, 2.13418449401187214349639258262, 3.25589680566919957042860187622, 3.95843649175983295343308045386, 4.87754967694813857178180932234, 5.73181894695275610502904723811, 6.60005625756178883466529574498, 7.34287298161908838889461920616, 7.77303004100725999372811788161, 9.020811866519435263649538136557

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