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Properties

Label 2-24e2-16.3-c2-0-4
Degree $2$
Conductor $576$
Sign $0.187 - 0.982i$
Analytic cond. $15.6948$
Root an. cond. $3.96167$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.68 + 3.68i)5-s + 9.66·7-s + (5.51 + 5.51i)11-s + (−6.27 − 6.27i)13-s + 6.78·17-s + (−13.5 + 13.5i)19-s + 17.0·23-s − 2.17i·25-s + (−4.85 − 4.85i)29-s + 5.25i·31-s + (−35.6 + 35.6i)35-s + (−18.1 + 18.1i)37-s + 48.2i·41-s + (54.5 + 54.5i)43-s + 40.4i·47-s + ⋯
L(s)  = 1  + (−0.737 + 0.737i)5-s + 1.38·7-s + (0.501 + 0.501i)11-s + (−0.482 − 0.482i)13-s + 0.399·17-s + (−0.711 + 0.711i)19-s + 0.742·23-s − 0.0868i·25-s + (−0.167 − 0.167i)29-s + 0.169i·31-s + (−1.01 + 1.01i)35-s + (−0.491 + 0.491i)37-s + 1.17i·41-s + (1.26 + 1.26i)43-s + 0.859i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.187 - 0.982i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.187 - 0.982i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.637975840\)
\(L(\frac12)\) \(\approx\) \(1.637975840\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (3.68 - 3.68i)T - 25iT^{2} \)
7 \( 1 - 9.66T + 49T^{2} \)
11 \( 1 + (-5.51 - 5.51i)T + 121iT^{2} \)
13 \( 1 + (6.27 + 6.27i)T + 169iT^{2} \)
17 \( 1 - 6.78T + 289T^{2} \)
19 \( 1 + (13.5 - 13.5i)T - 361iT^{2} \)
23 \( 1 - 17.0T + 529T^{2} \)
29 \( 1 + (4.85 + 4.85i)T + 841iT^{2} \)
31 \( 1 - 5.25iT - 961T^{2} \)
37 \( 1 + (18.1 - 18.1i)T - 1.36e3iT^{2} \)
41 \( 1 - 48.2iT - 1.68e3T^{2} \)
43 \( 1 + (-54.5 - 54.5i)T + 1.84e3iT^{2} \)
47 \( 1 - 40.4iT - 2.20e3T^{2} \)
53 \( 1 + (10.8 - 10.8i)T - 2.80e3iT^{2} \)
59 \( 1 + (-50.8 - 50.8i)T + 3.48e3iT^{2} \)
61 \( 1 + (17.0 + 17.0i)T + 3.72e3iT^{2} \)
67 \( 1 + (22.9 - 22.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 51.6T + 5.04e3T^{2} \)
73 \( 1 - 78.5iT - 5.32e3T^{2} \)
79 \( 1 + 108. iT - 6.24e3T^{2} \)
83 \( 1 + (-57.3 + 57.3i)T - 6.88e3iT^{2} \)
89 \( 1 + 44.1iT - 7.92e3T^{2} \)
97 \( 1 - 112.T + 9.40e3T^{2} \)
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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

−10.85016232893639953714499398789, −9.998715557750502524160069393821, −8.826296616531854326909854251896, −7.81887237548575992819322316389, −7.42538455497434117507513538543, −6.23053076636554308120532479766, −4.97704489869970826476804359559, −4.13595981349149219191112291941, −2.88147722675038619022767933422, −1.43800029808418196508340236115, 0.68243680930265674359099918153, 2.08017945133078048533965223085, 3.81911110561113006964152168881, 4.66504194688406236478617030232, 5.45314112127657808935390665067, 6.89809342109100151939595791324, 7.75899571403378922173386553674, 8.636825267037673041623855455711, 9.096912634405448344622770639032, 10.55053045649438179329479643849

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