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Properties

Label 2-24e2-9.5-c2-0-0
Degree $2$
Conductor $576$
Sign $-0.936 + 0.350i$
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  + (−2.44 + 1.73i)3-s + (4.5 + 2.59i)5-s + (−4.17 − 7.22i)7-s + (2.99 − 8.48i)9-s + (0.825 − 0.476i)11-s + (−4.84 + 8.39i)13-s + (−15.5 + 1.43i)15-s + 18.8i·17-s − 24.6·19-s + (22.7 + 10.4i)21-s + (−0.825 − 0.476i)23-s + (1 + 1.73i)25-s + (7.34 + 25.9i)27-s + (−11.8 + 6.84i)29-s + (1.52 − 2.63i)31-s + ⋯
L(s)  = 1  + (−0.816 + 0.577i)3-s + (0.900 + 0.519i)5-s + (−0.596 − 1.03i)7-s + (0.333 − 0.942i)9-s + (0.0750 − 0.0433i)11-s + (−0.372 + 0.645i)13-s + (−1.03 + 0.0953i)15-s + 1.11i·17-s − 1.29·19-s + (1.08 + 0.499i)21-s + (−0.0359 − 0.0207i)23-s + (0.0400 + 0.0692i)25-s + (0.272 + 0.962i)27-s + (−0.408 + 0.235i)29-s + (0.0491 − 0.0850i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.936 + 0.350i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ -0.936 + 0.350i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.08585026330\)
\(L(\frac12)\) \(\approx\) \(0.08585026330\)
\(L(2)\) 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 \( 1 + (2.44 - 1.73i)T \)
good5 \( 1 + (-4.5 - 2.59i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (4.17 + 7.22i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-0.825 + 0.476i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (4.84 - 8.39i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 18.8iT - 289T^{2} \)
19 \( 1 + 24.6T + 361T^{2} \)
23 \( 1 + (0.825 + 0.476i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (11.8 - 6.84i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-1.52 + 2.63i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 46.6T + 1.36e3T^{2} \)
41 \( 1 + (9.45 + 5.45i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (22.5 + 39.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (39.2 - 22.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 94.3iT - 2.80e3T^{2} \)
59 \( 1 + (16.2 + 9.39i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-6.54 - 11.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (37.5 - 64.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 18.0iT - 5.04e3T^{2} \)
73 \( 1 + 7.90T + 5.32e3T^{2} \)
79 \( 1 + (21.8 + 37.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (112. - 65.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 145. iT - 7.92e3T^{2} \)
97 \( 1 + (-54.9 - 95.1i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.64078180336186869330717090889, −10.31115901567874109409971776897, −9.619869152809187347491996460022, −8.539229973247792457739783881611, −6.94023902578078002041199752490, −6.55067386363148248894577142827, −5.62391994042219235850222016039, −4.38182885062435956841225093432, −3.54244097487525556179003260428, −1.83076158395789606916358366616, 0.03397242950084387879984588801, 1.71357653967474329733869377495, 2.79502802025449987847132248846, 4.74302383927982544278648517969, 5.55500507835713216341604628577, 6.18174945966648314597604852453, 7.12655971716861011197954243094, 8.315333958530969617748960974804, 9.255762658809239901524647497874, 9.978250288863614250943042614776

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