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Properties

Label 2-3024-63.47-c1-0-4
Degree $2$
Conductor $3024$
Sign $-0.398 - 0.916i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.11·5-s + (2.54 + 0.711i)7-s − 5.82i·11-s + (−2.52 + 1.45i)13-s + (−1.58 − 2.73i)17-s + (−0.722 − 0.417i)19-s + 7.09i·23-s + 11.9·25-s + (1.91 + 1.10i)29-s + (−3.66 − 2.11i)31-s + (−10.4 − 2.92i)35-s + (1.82 − 3.16i)37-s + (−2.04 − 3.54i)41-s + (−0.155 + 0.269i)43-s + (0.502 + 0.870i)47-s + ⋯
L(s)  = 1  − 1.84·5-s + (0.963 + 0.269i)7-s − 1.75i·11-s + (−0.699 + 0.403i)13-s + (−0.383 − 0.664i)17-s + (−0.165 − 0.0956i)19-s + 1.47i·23-s + 2.38·25-s + (0.355 + 0.205i)29-s + (−0.658 − 0.380i)31-s + (−1.77 − 0.495i)35-s + (0.300 − 0.519i)37-s + (−0.319 − 0.554i)41-s + (−0.0237 + 0.0410i)43-s + (0.0732 + 0.126i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.398 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.398 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.398 - 0.916i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.398 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4605236891\)
\(L(\frac12)\) \(\approx\) \(0.4605236891\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-2.54 - 0.711i)T \)
good5 \( 1 + 4.11T + 5T^{2} \)
11 \( 1 + 5.82iT - 11T^{2} \)
13 \( 1 + (2.52 - 1.45i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.58 + 2.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.722 + 0.417i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.09iT - 23T^{2} \)
29 \( 1 + (-1.91 - 1.10i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.66 + 2.11i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.82 + 3.16i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.04 + 3.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.155 - 0.269i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.502 - 0.870i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.94 + 1.12i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.51 + 4.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.98 - 2.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.99 - 8.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (3.04 - 1.76i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.579 + 1.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.57 - 13.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.82 - 8.35i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.06 - 2.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.646646394695593312794896589000, −8.297229130024154655619361681731, −7.46601450003855933668153237639, −7.05679662316740760516738532441, −5.76163817120516630758692569679, −5.04786034607712655390683870096, −4.17497581850362064424558347192, −3.52400013964259729047806954241, −2.57326568663058107517857058164, −1.01568438011772817582052209578, 0.17481200056625926873085445424, 1.68425226320948362827252080538, 2.81844091617901383972691491784, 4.01088854858042305487984801218, 4.56016764068673762341603451664, 4.94001823253623833658858410818, 6.47123104828982285722380029663, 7.28061385684803506733921131009, 7.66819666603687435078888449250, 8.312561666549460688568575877908

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