Properties

Degree $2$
Conductor $576$
Sign $0.973 + 0.229i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.00 + 1.00i)5-s − 10.0·7-s + (2.26 + 2.26i)11-s + (−6.88 − 6.88i)13-s + 22.3·17-s + (16.8 − 16.8i)19-s + 33.2·23-s + 22.9i·25-s + (24.6 + 24.6i)29-s − 41.3i·31-s + (10.1 − 10.1i)35-s + (−6.60 + 6.60i)37-s − 47.1i·41-s + (48.8 + 48.8i)43-s − 45.6i·47-s + ⋯
L(s)  = 1  + (−0.201 + 0.201i)5-s − 1.43·7-s + (0.205 + 0.205i)11-s + (−0.529 − 0.529i)13-s + 1.31·17-s + (0.889 − 0.889i)19-s + 1.44·23-s + 0.918i·25-s + (0.849 + 0.849i)29-s − 1.33i·31-s + (0.288 − 0.288i)35-s + (−0.178 + 0.178i)37-s − 1.14i·41-s + (1.13 + 1.13i)43-s − 0.970i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.973 + 0.229i$
Motivic weight: \(2\)
Character: $\chi_{576} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.973 + 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.427851995\)
\(L(\frac12)\) \(\approx\) \(1.427851995\)
\(L(2)\) 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 \( 1 + (1.00 - 1.00i)T - 25iT^{2} \)
7 \( 1 + 10.0T + 49T^{2} \)
11 \( 1 + (-2.26 - 2.26i)T + 121iT^{2} \)
13 \( 1 + (6.88 + 6.88i)T + 169iT^{2} \)
17 \( 1 - 22.3T + 289T^{2} \)
19 \( 1 + (-16.8 + 16.8i)T - 361iT^{2} \)
23 \( 1 - 33.2T + 529T^{2} \)
29 \( 1 + (-24.6 - 24.6i)T + 841iT^{2} \)
31 \( 1 + 41.3iT - 961T^{2} \)
37 \( 1 + (6.60 - 6.60i)T - 1.36e3iT^{2} \)
41 \( 1 + 47.1iT - 1.68e3T^{2} \)
43 \( 1 + (-48.8 - 48.8i)T + 1.84e3iT^{2} \)
47 \( 1 + 45.6iT - 2.20e3T^{2} \)
53 \( 1 + (25.1 - 25.1i)T - 2.80e3iT^{2} \)
59 \( 1 + (-6.23 - 6.23i)T + 3.48e3iT^{2} \)
61 \( 1 + (-35.9 - 35.9i)T + 3.72e3iT^{2} \)
67 \( 1 + (10.2 - 10.2i)T - 4.48e3iT^{2} \)
71 \( 1 - 11.9T + 5.04e3T^{2} \)
73 \( 1 + 111. iT - 5.32e3T^{2} \)
79 \( 1 - 4.46iT - 6.24e3T^{2} \)
83 \( 1 + (-10.1 + 10.1i)T - 6.88e3iT^{2} \)
89 \( 1 + 21.9iT - 7.92e3T^{2} \)
97 \( 1 - 107.T + 9.40e3T^{2} \)
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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

−10.34667621167723278036982996305, −9.610659178472139435087766482633, −8.974362569728312302314326859645, −7.52813699702107237248498752563, −7.05172069979766015379476499597, −5.92383258850872945892771295342, −4.96809542932643923344988029514, −3.46347480290923261585538150213, −2.85241758109480438756523406667, −0.77301667450612960546936123298, 0.929438981746847921708529555287, 2.84279104396646476927322929669, 3.67878456345339410246198596199, 4.98661076692545840467099160834, 6.07484332428021365530010015773, 6.89801643224515567500452774292, 7.84214523953820885980981056025, 8.929688250598287900775133878789, 9.755063235043961671100785626238, 10.28174557398174502985333385860

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