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Properties

Label 2-3024-63.4-c1-0-3
Degree $2$
Conductor $3024$
Sign $-0.609 - 0.792i$
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  + 0.460·5-s + (−2.25 − 1.38i)7-s − 3.64·11-s + (0.730 − 1.26i)13-s + (1.86 − 3.23i)17-s + (2.02 + 3.51i)19-s + 1.13·23-s − 4.78·25-s + (4.48 + 7.77i)29-s + (−0.257 − 0.445i)31-s + (−1.03 − 0.635i)35-s + (−4.55 − 7.88i)37-s + (0.472 − 0.819i)41-s + (−4.66 − 8.07i)43-s + (−1.16 + 2.01i)47-s + ⋯
L(s)  = 1  + 0.205·5-s + (−0.853 − 0.521i)7-s − 1.09·11-s + (0.202 − 0.350i)13-s + (0.452 − 0.784i)17-s + (0.465 + 0.805i)19-s + 0.236·23-s − 0.957·25-s + (0.833 + 1.44i)29-s + (−0.0462 − 0.0800i)31-s + (−0.175 − 0.107i)35-s + (−0.748 − 1.29i)37-s + (0.0738 − 0.127i)41-s + (−0.711 − 1.23i)43-s + (−0.169 + 0.294i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.609 - 0.792i$
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: \((2,\ 3024,\ (\ :1/2),\ -0.609 - 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4552486106\)
\(L(\frac12)\) \(\approx\) \(0.4552486106\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.25 + 1.38i)T \)
good5 \( 1 - 0.460T + 5T^{2} \)
11 \( 1 + 3.64T + 11T^{2} \)
13 \( 1 + (-0.730 + 1.26i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.86 + 3.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.02 - 3.51i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.13T + 23T^{2} \)
29 \( 1 + (-4.48 - 7.77i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.257 + 0.445i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.55 + 7.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.472 + 0.819i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.66 + 8.07i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.16 - 2.01i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.21 - 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.44 - 11.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.04 - 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.16 + 2.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.67T + 71T^{2} \)
73 \( 1 + (6.62 - 11.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.50 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.32 - 5.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.36 - 2.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.59 + 9.68i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.062828886180036817343717418682, −8.174605029530048031191123872564, −7.38420585320662590709302165166, −6.92040205480395769520891085120, −5.71155202063568135377196470368, −5.42844551488053269624914550585, −4.22427347013221353067744254034, −3.31203280041649923699847922402, −2.63596472266839259837896820871, −1.21422868410273807083826324519, 0.14532489233371524162061090627, 1.78616860756131160408407664433, 2.80793402764776773489161329333, 3.48632562180316368901437851730, 4.68769053765314830911985642778, 5.40706230138942106236518013658, 6.27352548069771236024897289030, 6.73565581165847714791071652251, 7.961467511397161469978292366930, 8.266298601459017641135541278521

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