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Properties

Label 2-24e2-144.11-c1-0-6
Degree $2$
Conductor $576$
Sign $0.369 - 0.929i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.11 + 1.32i)3-s + (1.20 + 0.323i)5-s + (−0.140 + 0.242i)7-s + (−0.513 − 2.95i)9-s + (3.07 − 0.823i)11-s + (2.76 + 0.740i)13-s + (−1.77 + 1.23i)15-s + 3.72i·17-s + (4.10 + 4.10i)19-s + (−0.165 − 0.456i)21-s + (1.57 − 0.909i)23-s + (−2.98 − 1.72i)25-s + (4.49 + 2.61i)27-s + (−3.83 + 1.02i)29-s + (−8.81 + 5.08i)31-s + ⋯
L(s)  = 1  + (−0.643 + 0.765i)3-s + (0.539 + 0.144i)5-s + (−0.0530 + 0.0918i)7-s + (−0.171 − 0.985i)9-s + (0.926 − 0.248i)11-s + (0.766 + 0.205i)13-s + (−0.457 + 0.319i)15-s + 0.902i·17-s + (0.942 + 0.942i)19-s + (−0.0361 − 0.0996i)21-s + (0.328 − 0.189i)23-s + (−0.596 − 0.344i)25-s + (0.864 + 0.503i)27-s + (−0.711 + 0.190i)29-s + (−1.58 + 0.913i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.369 - 0.929i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.369 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11174 + 0.754771i\)
\(L(\frac12)\) \(\approx\) \(1.11174 + 0.754771i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.11 - 1.32i)T \)
good5 \( 1 + (-1.20 - 0.323i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.140 - 0.242i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.07 + 0.823i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-2.76 - 0.740i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 3.72iT - 17T^{2} \)
19 \( 1 + (-4.10 - 4.10i)T + 19iT^{2} \)
23 \( 1 + (-1.57 + 0.909i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.83 - 1.02i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (8.81 - 5.08i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.76 - 1.76i)T + 37iT^{2} \)
41 \( 1 + (-2.66 - 4.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.84 - 6.89i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-5.48 + 9.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.58 + 8.58i)T - 53iT^{2} \)
59 \( 1 + (1.44 - 5.38i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.66 - 6.23i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.00 + 3.75i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.6iT - 71T^{2} \)
73 \( 1 - 9.30iT - 73T^{2} \)
79 \( 1 + (8.70 + 5.02i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.588 + 2.19i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 1.87T + 89T^{2} \)
97 \( 1 + (-9.19 + 15.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.85274657518005627582458296332, −10.05523050024754055975641793095, −9.280252580036796585116189977306, −8.528088430804676224409386045632, −7.09756763232792781486697260308, −6.02652872191770861961034240447, −5.62588526867644200629821240439, −4.18083472620377397994978132912, −3.40794574558464972664560006548, −1.46829104154774207949665212892, 0.959553113189505594578019459901, 2.27572167016183980376945691848, 3.89039608523382172975257573151, 5.31186063325605231608666303345, 5.89444511175704646516946110681, 7.04433607984727653196466208329, 7.54776876615729676779358972991, 9.003792466141175124510773980474, 9.515265749039820520368791514462, 10.83005410491589766011894434616

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