Properties

Degree $2$
Conductor $288$
Sign $-0.104 + 0.994i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.20 − 1.59i)2-s + (−1.09 + 3.84i)4-s + (0.642 + 1.55i)5-s + (−4.95 − 4.95i)7-s + (7.46 − 2.88i)8-s + (1.70 − 2.89i)10-s + (4.27 + 10.3i)11-s + (1.68 − 4.06i)13-s + (−1.93 + 13.8i)14-s + (−13.6 − 8.42i)16-s − 28.6i·17-s + (17.5 + 7.26i)19-s + (−6.67 + 0.773i)20-s + (11.3 − 19.2i)22-s + (24.3 − 24.3i)23-s + ⋯
L(s)  = 1  + (−0.602 − 0.798i)2-s + (−0.273 + 0.961i)4-s + (0.128 + 0.310i)5-s + (−0.707 − 0.707i)7-s + (0.932 − 0.361i)8-s + (0.170 − 0.289i)10-s + (0.388 + 0.937i)11-s + (0.129 − 0.312i)13-s + (−0.138 + 0.990i)14-s + (−0.850 − 0.526i)16-s − 1.68i·17-s + (0.923 + 0.382i)19-s + (−0.333 + 0.0386i)20-s + (0.514 − 0.874i)22-s + (1.05 − 1.05i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.104 + 0.994i$
Motivic weight: \(2\)
Character: $\chi_{288} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ -0.104 + 0.994i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.686751 - 0.762450i\)
\(L(\frac12)\) \(\approx\) \(0.686751 - 0.762450i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 + 1.59i)T \)
3 \( 1 \)
good5 \( 1 + (-0.642 - 1.55i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (4.95 + 4.95i)T + 49iT^{2} \)
11 \( 1 + (-4.27 - 10.3i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (-1.68 + 4.06i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 28.6iT - 289T^{2} \)
19 \( 1 + (-17.5 - 7.26i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (-24.3 + 24.3i)T - 529iT^{2} \)
29 \( 1 + (8.57 + 3.55i)T + (594. + 594. i)T^{2} \)
31 \( 1 - 5.73iT - 961T^{2} \)
37 \( 1 + (26.1 + 63.0i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (-14.2 - 14.2i)T + 1.68e3iT^{2} \)
43 \( 1 + (10.1 + 24.4i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 57.9T + 2.20e3T^{2} \)
53 \( 1 + (-46.3 + 19.2i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-27.6 + 11.4i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-76.3 - 31.6i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (36.1 - 87.3i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-5.39 - 5.39i)T + 5.04e3iT^{2} \)
73 \( 1 + (25.4 + 25.4i)T + 5.32e3iT^{2} \)
79 \( 1 - 50.1T + 6.24e3T^{2} \)
83 \( 1 + (-100. - 41.7i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (10.6 - 10.6i)T - 7.92e3iT^{2} \)
97 \( 1 + 14.3T + 9.40e3T^{2} \)
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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

−11.23196065563683790866279854928, −10.25348055249777189925014772195, −9.691684968762002060753605803215, −8.760105131029076561256952012388, −7.31509960001272641522480362329, −6.89221873310406569014820946189, −4.95701933140879993148126456666, −3.66908349280055161142397087129, −2.54721669611721859599384957080, −0.71138039526716568891071374603, 1.33881544054435261930060410942, 3.39968769328286142410989002323, 5.13168043411339216577580202889, 6.01171760326828089541358362225, 6.85188252775060876978387799341, 8.168074832064291225207502730534, 8.975833586594229641297694598323, 9.577607301209660782926562828279, 10.76277815133685994743135179268, 11.68047947147474579948721248621

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