Properties

Label 2-3024-252.67-c0-0-1
Degree $2$
Conductor $3024$
Sign $0.916 + 0.400i$
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  + 5-s + (0.866 + 0.5i)7-s i·11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s i·23-s + (−0.5 − 0.866i)29-s + (0.866 + 0.5i)35-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (−0.866 + 0.5i)43-s + (0.499 + 0.866i)49-s + (−0.5 + 0.866i)53-s i·55-s + ⋯
L(s)  = 1  + 5-s + (0.866 + 0.5i)7-s i·11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s i·23-s + (−0.5 − 0.866i)29-s + (0.866 + 0.5i)35-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + (−0.866 + 0.5i)43-s + (0.499 + 0.866i)49-s + (−0.5 + 0.866i)53-s i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.916 + 0.400i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (3007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ 0.916 + 0.400i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.651038423\)
\(L(\frac12)\) \(\approx\) \(1.651038423\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 - T + T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.737840460267887155699788413145, −8.239924773873139799000644763337, −7.52161872611361802735544279355, −6.21746511288709485677171394781, −5.91378962935762865518366610226, −5.18187938329127415530572722742, −4.24274434810113407003037279543, −3.04024375250871161177008710990, −2.27283431008472217328534827076, −1.14135811186556960384391995694, 1.71885085858526978628349529123, 1.86144454536806154846744435504, 3.45945826878210643824807223350, 4.38336218551121490118980430582, 5.04427850440481505862219094649, 5.97362816935965176699709774405, 6.63816175228364708981597958813, 7.48688767735395361767893634131, 8.141501913736016458559848430347, 9.149866576654530072848035755148

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