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Properties

Label 2-3024-63.4-c1-0-16
Degree $2$
Conductor $3024$
Sign $0.444 - 0.895i$
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  + 0.468·5-s + (2.39 + 1.13i)7-s − 1.34·11-s + (−3.16 + 5.48i)13-s + (2.47 − 4.28i)17-s + (−2.38 − 4.13i)19-s + 7.62·23-s − 4.78·25-s + (1.80 + 3.12i)29-s + (3.24 + 5.62i)31-s + (1.11 + 0.531i)35-s + (5.24 + 9.07i)37-s + (0.0251 − 0.0435i)41-s + (0.431 + 0.748i)43-s + (5.49 − 9.51i)47-s + ⋯
L(s)  = 1  + 0.209·5-s + (0.903 + 0.428i)7-s − 0.406·11-s + (−0.877 + 1.52i)13-s + (0.599 − 1.03i)17-s + (−0.548 − 0.949i)19-s + 1.59·23-s − 0.956·25-s + (0.335 + 0.580i)29-s + (0.583 + 1.01i)31-s + (0.189 + 0.0897i)35-s + (0.861 + 1.49i)37-s + (0.00392 − 0.00680i)41-s + (0.0658 + 0.114i)43-s + (0.801 − 1.38i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.444 - 0.895i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.444 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.888825120\)
\(L(\frac12)\) \(\approx\) \(1.888825120\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.39 - 1.13i)T \)
good5 \( 1 - 0.468T + 5T^{2} \)
11 \( 1 + 1.34T + 11T^{2} \)
13 \( 1 + (3.16 - 5.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.47 + 4.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.38 + 4.13i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.62T + 23T^{2} \)
29 \( 1 + (-1.80 - 3.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.24 - 5.62i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.24 - 9.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.0251 + 0.0435i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.431 - 0.748i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.49 + 9.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.84 - 10.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.93 - 3.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.87 - 3.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.32 + 2.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.04T + 71T^{2} \)
73 \( 1 + (3.30 - 5.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.58 + 2.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.90 - 8.49i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.30 + 9.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.97 - 12.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.989991254521495572729447951425, −8.147481650560097436373527111863, −7.21948346146374563391385555576, −6.81636480671822758566718260640, −5.68518188673123811302485224449, −4.76178949930766242888432796382, −4.60443610429164739585508314624, −2.99850687595699715291520546609, −2.31951786057126889015148479302, −1.19664874086911718960914718788, 0.64141324562200935107067983374, 1.87764221155903749243732404837, 2.84897303716229983834698714138, 3.90831137701700278028725664631, 4.74034476279317080002739224436, 5.58684186981102020771949730151, 6.09649282100148224231096102726, 7.39869241130942854376619875029, 7.86471424390369191743123628578, 8.296346969759437976044886972717

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