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Properties

Label 2-24e2-16.3-c4-0-23
Degree $2$
Conductor $576$
Sign $0.876 - 0.480i$
Analytic cond. $59.5410$
Root an. cond. $7.71628$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.84 − 2.84i)5-s + 76.7·7-s + (121. + 121. i)11-s + (27.1 + 27.1i)13-s + 88.0·17-s + (261. − 261. i)19-s + 93.4·23-s + 608. i·25-s + (−272. − 272. i)29-s + 1.23e3i·31-s + (218. − 218. i)35-s + (−1.04e3 + 1.04e3i)37-s − 915. i·41-s + (−1.11e3 − 1.11e3i)43-s − 1.72e3i·47-s + ⋯
L(s)  = 1  + (0.113 − 0.113i)5-s + 1.56·7-s + (1.00 + 1.00i)11-s + (0.160 + 0.160i)13-s + 0.304·17-s + (0.723 − 0.723i)19-s + 0.176·23-s + 0.974i·25-s + (−0.324 − 0.324i)29-s + 1.28i·31-s + (0.178 − 0.178i)35-s + (−0.764 + 0.764i)37-s − 0.544i·41-s + (−0.604 − 0.604i)43-s − 0.778i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.876 - 0.480i$
Analytic conductor: \(59.5410\)
Root analytic conductor: \(7.71628\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :2),\ 0.876 - 0.480i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.970025576\)
\(L(\frac12)\) \(\approx\) \(2.970025576\)
\(L(3)\) 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 \( 1 + (-2.84 + 2.84i)T - 625iT^{2} \)
7 \( 1 - 76.7T + 2.40e3T^{2} \)
11 \( 1 + (-121. - 121. i)T + 1.46e4iT^{2} \)
13 \( 1 + (-27.1 - 27.1i)T + 2.85e4iT^{2} \)
17 \( 1 - 88.0T + 8.35e4T^{2} \)
19 \( 1 + (-261. + 261. i)T - 1.30e5iT^{2} \)
23 \( 1 - 93.4T + 2.79e5T^{2} \)
29 \( 1 + (272. + 272. i)T + 7.07e5iT^{2} \)
31 \( 1 - 1.23e3iT - 9.23e5T^{2} \)
37 \( 1 + (1.04e3 - 1.04e3i)T - 1.87e6iT^{2} \)
41 \( 1 + 915. iT - 2.82e6T^{2} \)
43 \( 1 + (1.11e3 + 1.11e3i)T + 3.41e6iT^{2} \)
47 \( 1 + 1.72e3iT - 4.87e6T^{2} \)
53 \( 1 + (734. - 734. i)T - 7.89e6iT^{2} \)
59 \( 1 + (1.20e3 + 1.20e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (-580. - 580. i)T + 1.38e7iT^{2} \)
67 \( 1 + (-1.48e3 + 1.48e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 5.57e3T + 2.54e7T^{2} \)
73 \( 1 - 6.61e3iT - 2.83e7T^{2} \)
79 \( 1 + 5.39e3iT - 3.89e7T^{2} \)
83 \( 1 + (2.55e3 - 2.55e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.09e4iT - 6.27e7T^{2} \)
97 \( 1 - 4.71e3T + 8.85e7T^{2} \)
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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

−10.19419113666791577470762423721, −9.226578597296169450443295104545, −8.511732153857082733196955398595, −7.46014773223083205748803949545, −6.78822792934921685196701385854, −5.31081213004791551417863194488, −4.75111501309748909077661416992, −3.58368221201736715867698012008, −1.94885498304828431718170374604, −1.18948650005073076990809122462, 0.879618354513693512106693437909, 1.85872089157545597287973028940, 3.35925280909459650308225776118, 4.41085470214427168437531185985, 5.48054926196023357052630224476, 6.29970728872821269362623264677, 7.59284871543954180612239990341, 8.236168084774635393319842306712, 9.067367886720793574262711393543, 10.11292447176741804428568266984

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