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Properties

Label 2-24e2-36.11-c1-0-4
Degree $2$
Conductor $576$
Sign $0.642 - 0.766i$
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  − 1.73·3-s + (−1.5 + 0.866i)5-s + (−2.59 − 1.5i)7-s + 2.99·9-s + (2.59 − 4.5i)11-s + (0.5 + 0.866i)13-s + (2.59 − 1.49i)15-s + 3.46i·17-s + 6i·19-s + (4.5 + 2.59i)21-s + (2.59 + 4.5i)23-s + (−1 + 1.73i)25-s − 5.19·27-s + (7.5 + 4.33i)29-s + (−2.59 + 1.5i)31-s + ⋯
L(s)  = 1  − 1.00·3-s + (−0.670 + 0.387i)5-s + (−0.981 − 0.566i)7-s + 0.999·9-s + (0.783 − 1.35i)11-s + (0.138 + 0.240i)13-s + (0.670 − 0.387i)15-s + 0.840i·17-s + 1.37i·19-s + (0.981 + 0.566i)21-s + (0.541 + 0.938i)23-s + (−0.200 + 0.346i)25-s − 1.00·27-s + (1.39 + 0.804i)29-s + (−0.466 + 0.269i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.680973 + 0.317542i\)
\(L(\frac12)\) \(\approx\) \(0.680973 + 0.317542i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73T \)
good5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.59 + 1.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.59 + 4.5i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + (-2.59 - 4.5i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.5 - 4.33i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.59 - 1.5i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (-4.5 + 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.59 - 1.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.59 + 4.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 + (-2.59 - 4.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.79 - 4.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + (-12.9 - 7.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.59 - 4.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.46iT - 89T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.88715502606306856988162501915, −10.23313906763957442957144013539, −9.221278025421991369111567949673, −8.068506353126110813372620923471, −7.01894540865216022577403947926, −6.34445493060096628441858529601, −5.54493383256188284712420419318, −3.92382377081498783843180390507, −3.50333488228262660001855836690, −1.09778265675574754814669964318, 0.61443782988954891279061842451, 2.63239857125259955717652727454, 4.28119969767190887153147130362, 4.83151182124662210262477246137, 6.20155128365261271151956540827, 6.80925528625691606521091481284, 7.75021467344640633209296813093, 9.176979675726487667822759533133, 9.599342976697507645070229369174, 10.71157643938497939434727845130

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