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Properties

Label 2-3024-84.47-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: $\chi_{3024} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ 0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.310223537\)
\(L(\frac12)\) \(\approx\) \(1.310223537\)
\(L(1)\) 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

−8.809932987060644859281481921601, −8.423128233155485569584964790371, −7.48735618547122553543260943465, −6.84080001105122167768035089021, −5.95767403939859113111777881765, −5.15108184770572835663422759257, −4.25607902737805401880652112440, −3.69510532947841472793586242656, −2.15062846315576656132166300264, −1.58333164013750098284671519352, 0.848976058807316863234559188136, 2.24555312960624982864613379815, 3.03888374027195137818744422186, 4.21711518192820590699722009269, 4.99382874288788310386300440074, 5.56564702942171839125661740739, 6.63125115996096096557623452467, 7.30954734565115399472723505438, 8.304060813098640324011594380243, 8.470691771187747529619256644682

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