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Properties

Label 2-12e2-16.11-c4-0-1
Degree $2$
Conductor $144$
Sign $-0.445 + 0.895i$
Analytic cond. $14.8852$
Root an. cond. $3.85814$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.59 + 3.66i)2-s + (−10.9 − 11.6i)4-s + (14.6 + 14.6i)5-s − 24.0·7-s + (60.3 − 21.3i)8-s + (−76.8 + 30.2i)10-s + (−61.7 + 61.7i)11-s + (−37.5 + 37.5i)13-s + (38.2 − 88.1i)14-s + (−17.6 + 255. i)16-s − 96.8·17-s + (−156. − 156. i)19-s + (11.4 − 330. i)20-s + (−128. − 324. i)22-s − 959.·23-s + ⋯
L(s)  = 1  + (−0.398 + 0.917i)2-s + (−0.682 − 0.731i)4-s + (0.584 + 0.584i)5-s − 0.490·7-s + (0.942 − 0.334i)8-s + (−0.768 + 0.302i)10-s + (−0.510 + 0.510i)11-s + (−0.222 + 0.222i)13-s + (0.195 − 0.449i)14-s + (−0.0689 + 0.997i)16-s − 0.335·17-s + (−0.434 − 0.434i)19-s + (0.0285 − 0.825i)20-s + (−0.264 − 0.671i)22-s − 1.81·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.445 + 0.895i$
Analytic conductor: \(14.8852\)
Root analytic conductor: \(3.85814\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :2),\ -0.445 + 0.895i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0516855 - 0.0834520i\)
\(L(\frac12)\) \(\approx\) \(0.0516855 - 0.0834520i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.59 - 3.66i)T \)
3 \( 1 \)
good5 \( 1 + (-14.6 - 14.6i)T + 625iT^{2} \)
7 \( 1 + 24.0T + 2.40e3T^{2} \)
11 \( 1 + (61.7 - 61.7i)T - 1.46e4iT^{2} \)
13 \( 1 + (37.5 - 37.5i)T - 2.85e4iT^{2} \)
17 \( 1 + 96.8T + 8.35e4T^{2} \)
19 \( 1 + (156. + 156. i)T + 1.30e5iT^{2} \)
23 \( 1 + 959.T + 2.79e5T^{2} \)
29 \( 1 + (-350. + 350. i)T - 7.07e5iT^{2} \)
31 \( 1 + 237. iT - 9.23e5T^{2} \)
37 \( 1 + (560. + 560. i)T + 1.87e6iT^{2} \)
41 \( 1 + 1.80e3iT - 2.82e6T^{2} \)
43 \( 1 + (-206. + 206. i)T - 3.41e6iT^{2} \)
47 \( 1 + 1.59e3iT - 4.87e6T^{2} \)
53 \( 1 + (-2.23e3 - 2.23e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (2.35e3 - 2.35e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (4.44e3 - 4.44e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (3.99e3 + 3.99e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 4.92e3T + 2.54e7T^{2} \)
73 \( 1 - 2.65e3iT - 2.83e7T^{2} \)
79 \( 1 - 8.79e3iT - 3.89e7T^{2} \)
83 \( 1 + (-228. - 228. i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.05e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.10e4T + 8.85e7T^{2} \)
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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

−13.45407588856572436270636886985, −12.20908789755391901957667091083, −10.55290125120029277688636035940, −9.989242094920790157994469006173, −8.925581667097004057516077199388, −7.67034367189556987912040398793, −6.64425801700138507473139072715, −5.77772801421185832854202041610, −4.30680652366140566770832185534, −2.22409339070852393593260279311, 0.04258773369302950163090173301, 1.73432062376155469798870707688, 3.20536668140822338701977826809, 4.70761856189957202412182140224, 6.07496363793123329872357813182, 7.82312283912497528713365775316, 8.781572773278310870696147685108, 9.793602441771776064890298585520, 10.51887386451692101513566889827, 11.74788359170984458353865084997

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