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Properties

Label 2-24e2-24.5-c6-0-38
Degree $2$
Conductor $576$
Sign $-0.169 + 0.985i$
Analytic cond. $132.511$
Root an. cond. $11.5113$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 168.·5-s + 13.9·7-s − 1.50e3·11-s − 744. i·13-s + 2.96e3i·17-s + 6.31e3i·19-s − 9.99e3i·23-s + 1.29e4·25-s + 2.03e4·29-s − 3.92e4·31-s + 2.36e3·35-s − 8.92e4i·37-s − 2.00e4i·41-s − 5.54e4i·43-s − 1.08e5i·47-s + ⋯
L(s)  = 1  + 1.35·5-s + 0.0407·7-s − 1.12·11-s − 0.339i·13-s + 0.603i·17-s + 0.920i·19-s − 0.821i·23-s + 0.827·25-s + 0.836·29-s − 1.31·31-s + 0.0551·35-s − 1.76i·37-s − 0.291i·41-s − 0.697i·43-s − 1.04i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.169 + 0.985i$
Analytic conductor: \(132.511\)
Root analytic conductor: \(11.5113\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :3),\ -0.169 + 0.985i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.760187914\)
\(L(\frac12)\) \(\approx\) \(1.760187914\)
\(L(4)\) 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 \( 1 - 168.T + 1.56e4T^{2} \)
7 \( 1 - 13.9T + 1.17e5T^{2} \)
11 \( 1 + 1.50e3T + 1.77e6T^{2} \)
13 \( 1 + 744. iT - 4.82e6T^{2} \)
17 \( 1 - 2.96e3iT - 2.41e7T^{2} \)
19 \( 1 - 6.31e3iT - 4.70e7T^{2} \)
23 \( 1 + 9.99e3iT - 1.48e8T^{2} \)
29 \( 1 - 2.03e4T + 5.94e8T^{2} \)
31 \( 1 + 3.92e4T + 8.87e8T^{2} \)
37 \( 1 + 8.92e4iT - 2.56e9T^{2} \)
41 \( 1 + 2.00e4iT - 4.75e9T^{2} \)
43 \( 1 + 5.54e4iT - 6.32e9T^{2} \)
47 \( 1 + 1.08e5iT - 1.07e10T^{2} \)
53 \( 1 - 1.43e5T + 2.21e10T^{2} \)
59 \( 1 - 1.10e4T + 4.21e10T^{2} \)
61 \( 1 + 1.75e5iT - 5.15e10T^{2} \)
67 \( 1 - 4.20e4iT - 9.04e10T^{2} \)
71 \( 1 + 1.03e5iT - 1.28e11T^{2} \)
73 \( 1 - 5.39e5T + 1.51e11T^{2} \)
79 \( 1 - 4.79e5T + 2.43e11T^{2} \)
83 \( 1 + 4.85e5T + 3.26e11T^{2} \)
89 \( 1 - 3.10e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.99e5T + 8.32e11T^{2} \)
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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

−9.658341130894993211394851974707, −8.675345785091199422269342405843, −7.82653036633445296712359738057, −6.69606987550557817229076173422, −5.71136165905762396731789401581, −5.22484206078010023974153635868, −3.78772041572311548467284672547, −2.48480610259008347901394629872, −1.76696366527867342062516340680, −0.33071198769961301111048668104, 1.14916675964409051842946957597, 2.24899269980420448781234936841, 3.07002101275414379691044197352, 4.72588176635234248058164046085, 5.39801253843398224667498084262, 6.33876444129590937977030920955, 7.25792416006404393189566345893, 8.312481963945255780949859165701, 9.368383091988043849858521227872, 9.856264162499664599287042321863

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