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Properties

Label 2-24e2-16.3-c4-0-24
Degree $2$
Conductor $576$
Sign $0.225 + 0.974i$
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  + (−4.72 + 4.72i)5-s − 45.3·7-s + (110. + 110. i)11-s + (−157. − 157. i)13-s + 378.·17-s + (−203. + 203. i)19-s − 740.·23-s + 580. i·25-s + (−82.6 − 82.6i)29-s + 286. i·31-s + (214. − 214. i)35-s + (1.47e3 − 1.47e3i)37-s − 1.30e3i·41-s + (366. + 366. i)43-s − 751. i·47-s + ⋯
L(s)  = 1  + (−0.188 + 0.188i)5-s − 0.925·7-s + (0.910 + 0.910i)11-s + (−0.929 − 0.929i)13-s + 1.31·17-s + (−0.562 + 0.562i)19-s − 1.39·23-s + 0.928i·25-s + (−0.0982 − 0.0982i)29-s + 0.297i·31-s + (0.174 − 0.174i)35-s + (1.07 − 1.07i)37-s − 0.774i·41-s + (0.198 + 0.198i)43-s − 0.340i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.225 + 0.974i$
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: \((2,\ 576,\ (\ :2),\ 0.225 + 0.974i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.061753807\)
\(L(\frac12)\) \(\approx\) \(1.061753807\)
\(L(3)\) 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 \( 1 + (4.72 - 4.72i)T - 625iT^{2} \)
7 \( 1 + 45.3T + 2.40e3T^{2} \)
11 \( 1 + (-110. - 110. i)T + 1.46e4iT^{2} \)
13 \( 1 + (157. + 157. i)T + 2.85e4iT^{2} \)
17 \( 1 - 378.T + 8.35e4T^{2} \)
19 \( 1 + (203. - 203. i)T - 1.30e5iT^{2} \)
23 \( 1 + 740.T + 2.79e5T^{2} \)
29 \( 1 + (82.6 + 82.6i)T + 7.07e5iT^{2} \)
31 \( 1 - 286. iT - 9.23e5T^{2} \)
37 \( 1 + (-1.47e3 + 1.47e3i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.30e3iT - 2.82e6T^{2} \)
43 \( 1 + (-366. - 366. i)T + 3.41e6iT^{2} \)
47 \( 1 + 751. iT - 4.87e6T^{2} \)
53 \( 1 + (-1.92e3 + 1.92e3i)T - 7.89e6iT^{2} \)
59 \( 1 + (1.35e3 + 1.35e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (1.83e3 + 1.83e3i)T + 1.38e7iT^{2} \)
67 \( 1 + (2.20e3 - 2.20e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 8.97e3T + 2.54e7T^{2} \)
73 \( 1 + 9.35e3iT - 2.83e7T^{2} \)
79 \( 1 + 2.86e3iT - 3.89e7T^{2} \)
83 \( 1 + (1.03e3 - 1.03e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 5.17e3iT - 6.27e7T^{2} \)
97 \( 1 - 8.53e3T + 8.85e7T^{2} \)
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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

−9.863635263191594162530458943234, −9.355955131108352537653255450282, −7.967202747012763077984501435494, −7.32632250786499428695891117879, −6.31064545240673551587201076467, −5.41698802214634564314941518994, −4.08174310861230997832415520648, −3.23392103497819395469274565630, −1.90916268703626377412631055382, −0.32125659603437828877462597750, 0.943044506272253454078057289727, 2.50986729465784637696926180248, 3.66541763292589778995249605837, 4.56152676944701602246931419480, 5.98332941666831594136224636909, 6.54164296968348691889922179467, 7.67376166058703381474525427889, 8.610674591479525379497030381512, 9.559337053837489591590377461851, 10.05815027692461550486906569983

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