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Properties

Label 2-3024-9.4-c1-0-3
Degree $2$
Conductor $3024$
Sign $-0.839 - 0.542i$
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  + (−1.15 + 2.00i)5-s + (−0.5 − 0.866i)7-s + (−0.655 − 1.13i)11-s + (1.93 − 3.34i)13-s + 0.326·17-s − 3.08·19-s + (−1.81 + 3.15i)23-s + (−0.169 − 0.293i)25-s + (4.75 + 8.22i)29-s + (3.24 − 5.62i)31-s + 2.31·35-s − 2.31·37-s + (−4.74 + 8.22i)41-s + (0.0493 + 0.0855i)43-s + (−0.108 − 0.187i)47-s + ⋯
L(s)  = 1  + (−0.516 + 0.894i)5-s + (−0.188 − 0.327i)7-s + (−0.197 − 0.342i)11-s + (0.536 − 0.928i)13-s + 0.0792·17-s − 0.707·19-s + (−0.379 + 0.656i)23-s + (−0.0338 − 0.0586i)25-s + (0.882 + 1.52i)29-s + (0.583 − 1.01i)31-s + 0.390·35-s − 0.379·37-s + (−0.741 + 1.28i)41-s + (0.00753 + 0.0130i)43-s + (−0.0158 − 0.0273i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.839 - 0.542i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.839 - 0.542i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6046355090\)
\(L(\frac12)\) \(\approx\) \(0.6046355090\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (1.15 - 2.00i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.655 + 1.13i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.93 + 3.34i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.326T + 17T^{2} \)
19 \( 1 + 3.08T + 19T^{2} \)
23 \( 1 + (1.81 - 3.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.75 - 8.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.24 + 5.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.31T + 37T^{2} \)
41 \( 1 + (4.74 - 8.22i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.0493 - 0.0855i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.108 + 0.187i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 13.7T + 53T^{2} \)
59 \( 1 + (-1.22 + 2.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.68 - 13.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.93 - 5.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.77T + 71T^{2} \)
73 \( 1 + 5.99T + 73T^{2} \)
79 \( 1 + (7.29 + 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.04 - 5.26i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.52T + 89T^{2} \)
97 \( 1 + (-2.99 - 5.18i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.886014440223763474495168846067, −8.173079871583188259220750683831, −7.57437411788711105227273495372, −6.76129813144545387380342360651, −6.14324303342497493787914254697, −5.23714276245404677874957894883, −4.20432650139539775949758505938, −3.32144030920750393490579773172, −2.82127141685893433827236661382, −1.29862791247995905687933834215, 0.19730834415011534258779813436, 1.57637790830242295760224577436, 2.61172935317712770268397436363, 3.85287097643391285984138407141, 4.48872607173106371294924727933, 5.17207455122000540383263153564, 6.29733847928780137723011544664, 6.74333112481960022274927815198, 7.921845367188137718967669864673, 8.444564205651587230578152332914

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