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Properties

Label 2-3024-63.59-c1-0-14
Degree $2$
Conductor $3024$
Sign $0.980 - 0.198i$
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  − 2.18·5-s + (−2.64 − 0.0736i)7-s + 1.46i·11-s + (−2.92 − 1.69i)13-s + (1.32 − 2.28i)17-s + (−6.87 + 3.97i)19-s − 4.00i·23-s − 0.234·25-s + (6.71 − 3.87i)29-s + (−0.612 + 0.353i)31-s + (5.77 + 0.160i)35-s + (1.41 + 2.45i)37-s + (3.74 − 6.48i)41-s + (1.27 + 2.20i)43-s + (−6.27 + 10.8i)47-s + ⋯
L(s)  = 1  − 0.976·5-s + (−0.999 − 0.0278i)7-s + 0.441i·11-s + (−0.811 − 0.468i)13-s + (0.320 − 0.555i)17-s + (−1.57 + 0.911i)19-s − 0.836i·23-s − 0.0468·25-s + (1.24 − 0.719i)29-s + (−0.109 + 0.0634i)31-s + (0.975 + 0.0271i)35-s + (0.233 + 0.403i)37-s + (0.584 − 1.01i)41-s + (0.193 + 0.335i)43-s + (−0.915 + 1.58i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.980 - 0.198i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.980 - 0.198i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8075668835\)
\(L(\frac12)\) \(\approx\) \(0.8075668835\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 + 0.0736i)T \)
good5 \( 1 + 2.18T + 5T^{2} \)
11 \( 1 - 1.46iT - 11T^{2} \)
13 \( 1 + (2.92 + 1.69i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.32 + 2.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.87 - 3.97i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.00iT - 23T^{2} \)
29 \( 1 + (-6.71 + 3.87i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.612 - 0.353i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.41 - 2.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.74 + 6.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.27 - 2.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.27 - 10.8i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.41 + 1.39i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.71 - 11.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.75 - 3.89i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.92 - 5.05i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 + (3.95 + 2.28i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.69 + 8.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.70 + 2.95i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.61 - 8.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.38 - 3.68i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.622083207695872149803231802306, −7.962976029865757716338877125408, −7.31006578505483130537435428645, −6.52287121626230486956596176508, −5.83827414088610508890781533000, −4.61948760889139281207973526292, −4.12144699630336392921872073578, −3.11126365454571942329833652392, −2.31190696370053282882396098510, −0.55993118240390568479681132755, 0.48205293292912066341977289954, 2.16279668597333990784095703473, 3.18719437144276047886668986317, 3.90833491673216278607669606913, 4.68399807578474527628343155555, 5.69203859679536275530187298589, 6.68517131283024471143289816323, 6.99835528525795621938842801722, 8.094905058777551824981758570627, 8.549297815604553202540946857985

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