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Properties

Label 2-3024-63.20-c1-0-18
Degree $2$
Conductor $3024$
Sign $0.851 - 0.524i$
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  + (0.0868 + 0.150i)5-s + (2.60 + 0.486i)7-s + (3.25 + 1.87i)11-s + (−3.54 + 2.04i)13-s + 6.00·17-s − 6.26i·19-s + (−6.37 + 3.68i)23-s + (2.48 − 4.30i)25-s + (8.58 + 4.95i)29-s + (−3.76 + 2.17i)31-s + (0.152 + 0.433i)35-s + 7.23·37-s + (0.489 + 0.848i)41-s + (−0.0468 + 0.0811i)43-s + (−1.86 + 3.22i)47-s + ⋯
L(s)  = 1  + (0.0388 + 0.0672i)5-s + (0.982 + 0.184i)7-s + (0.980 + 0.566i)11-s + (−0.982 + 0.567i)13-s + 1.45·17-s − 1.43i·19-s + (−1.32 + 0.767i)23-s + (0.496 − 0.860i)25-s + (1.59 + 0.920i)29-s + (−0.676 + 0.390i)31-s + (0.0257 + 0.0732i)35-s + 1.18·37-s + (0.0765 + 0.132i)41-s + (−0.00714 + 0.0123i)43-s + (−0.271 + 0.470i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.851 - 0.524i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.851 - 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.247294431\)
\(L(\frac12)\) \(\approx\) \(2.247294431\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 - 0.486i)T \)
good5 \( 1 + (-0.0868 - 0.150i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.25 - 1.87i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.54 - 2.04i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.00T + 17T^{2} \)
19 \( 1 + 6.26iT - 19T^{2} \)
23 \( 1 + (6.37 - 3.68i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-8.58 - 4.95i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.76 - 2.17i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.23T + 37T^{2} \)
41 \( 1 + (-0.489 - 0.848i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.0468 - 0.0811i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.86 - 3.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.61iT - 53T^{2} \)
59 \( 1 + (-0.620 - 1.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.45 + 4.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.21 + 7.30i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 - 15.4iT - 73T^{2} \)
79 \( 1 + (-1.21 + 2.10i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.16 + 5.47i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + (-5.02 - 2.90i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.772345916747306683764584809274, −8.025907811188345315170466789613, −7.27250310727860142550304555683, −6.66635697463566648852281692507, −5.66978168524047299791966584699, −4.77555523805337873142062385397, −4.33393511980034025889833200115, −3.08451543914563120467899245940, −2.09406838952321559196613060093, −1.11619022990265062072068415075, 0.857562841327060150667044903072, 1.84295108771635792312524174321, 3.04473712441587698653316311489, 3.98751981363001395921611795703, 4.72905550461278675253000263943, 5.71716520041152118007817091227, 6.17244963017645368851916201810, 7.42288114536348377118397174689, 7.900073835477929036739921330784, 8.466100127866270770117170051601

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