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Properties

Label 2-3024-84.59-c0-0-1
Degree $2$
Conductor $3024$
Sign $0.605 + 0.795i$
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  − 7-s − 1.73i·13-s + (1 + 1.73i)19-s + (0.5 − 0.866i)25-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s − 1.73i·43-s + 49-s + (1.5 − 0.866i)61-s + (−1.5 − 0.866i)67-s + (1.5 − 0.866i)79-s + 1.73i·91-s + 1.73i·97-s + (0.5 + 0.866i)103-s + (0.5 − 0.866i)109-s + ⋯
L(s)  = 1  − 7-s − 1.73i·13-s + (1 + 1.73i)19-s + (0.5 − 0.866i)25-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s − 1.73i·43-s + 49-s + (1.5 − 0.866i)61-s + (−1.5 − 0.866i)67-s + (1.5 − 0.866i)79-s + 1.73i·91-s + 1.73i·97-s + (0.5 + 0.866i)103-s + (0.5 − 0.866i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.605 + 0.795i$
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: \((2,\ 3024,\ (\ :0),\ 0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.040998022\)
\(L(\frac12)\) \(\approx\) \(1.040998022\)
\(L(1)\) 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 \( 1 + T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + 1.73iT - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
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

−8.776972405607692961498789531765, −7.949497268168968776778115442177, −7.44581786522407746559651702457, −6.40173583459876422430965109846, −5.78013374214421030944250611222, −5.13228111935836079284942763387, −3.79486441903679941222616225541, −3.30580490963764900561089533974, −2.28447647272153658522705068103, −0.71597861909949129954474183164, 1.26955899974993790443636464035, 2.62716491819529291105978448016, 3.32940879611635655897020657269, 4.39990262138675167545200995909, 5.07106696181931412340056314717, 6.13543007974589541612230947422, 6.97267876167171196689573706985, 7.10559930711469195941974908517, 8.449167187839648686858355376303, 9.174562861719668583081566056735

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