Properties

Label 2-12e2-9.2-c2-0-1
Degree $2$
Conductor $144$
Sign $0.112 - 0.993i$
Analytic cond. $3.92371$
Root an. cond. $1.98083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.18 − 2.05i)3-s + (−2.05 + 1.18i)5-s + (−4.05 + 7.02i)7-s + (0.558 + 8.98i)9-s + (17.6 + 10.1i)11-s + (−3.05 − 5.29i)13-s + (6.94 + 1.63i)15-s + 17.9i·17-s − 9.11·19-s + (23.3 − 7.02i)21-s + (−29.0 + 16.7i)23-s + (−9.67 + 16.7i)25-s + (17.2 − 20.7i)27-s + (14.4 + 8.31i)29-s + (−11.1 − 19.3i)31-s + ⋯
L(s)  = 1  + (−0.728 − 0.684i)3-s + (−0.411 + 0.237i)5-s + (−0.579 + 1.00i)7-s + (0.0620 + 0.998i)9-s + (1.60 + 0.924i)11-s + (−0.235 − 0.407i)13-s + (0.462 + 0.108i)15-s + 1.05i·17-s − 0.479·19-s + (1.11 − 0.334i)21-s + (−1.26 + 0.729i)23-s + (−0.387 + 0.670i)25-s + (0.638 − 0.769i)27-s + (0.496 + 0.286i)29-s + (−0.360 − 0.624i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.112 - 0.993i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.112 - 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.568305 + 0.507743i\)
\(L(\frac12)\) \(\approx\) \(0.568305 + 0.507743i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (2.18 + 2.05i)T \)
good5 \( 1 + (2.05 - 1.18i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (4.05 - 7.02i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-17.6 - 10.1i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (3.05 + 5.29i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 17.9iT - 289T^{2} \)
19 \( 1 + 9.11T + 361T^{2} \)
23 \( 1 + (29.0 - 16.7i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-14.4 - 8.31i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (11.1 + 19.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 50.4T + 1.36e3T^{2} \)
41 \( 1 + (-29.9 + 17.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-11.5 + 19.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-33.1 - 19.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 19.0iT - 2.80e3T^{2} \)
59 \( 1 + (-2.96 + 1.71i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-23.1 + 40.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (3.14 + 5.45i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 35.9iT - 5.04e3T^{2} \)
73 \( 1 - 47.3T + 5.32e3T^{2} \)
79 \( 1 + (42.2 - 73.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-33.1 - 19.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 143. iT - 7.92e3T^{2} \)
97 \( 1 + (40.3 - 69.9i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.60886187896907820310796944689, −12.25953643838186336360235939480, −11.36433871993597297799655450351, −10.09888819913574581995941859312, −8.933471970740367358970774834565, −7.60465809793590322586196620587, −6.53934082316317018217289894828, −5.66898602710482737840393133874, −3.94914786953374667369238860954, −1.92157986650593612286107693687, 0.55238639663421763721218855534, 3.67560794279252365039610890123, 4.45635333501527627130821515314, 6.14078357180180067412212514364, 6.97045443707983262170098715538, 8.654328719676497497899593775973, 9.665867658030573891784873539167, 10.60717258771064680202038720861, 11.67158242049134946075826658422, 12.26334760966394651482480026545

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