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Properties

Label 2-12e2-9.5-c2-0-4
Degree $2$
Conductor $144$
Sign $0.984 - 0.173i$
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  + 3·3-s + (−3.39 − 1.96i)5-s + (6.39 + 11.0i)7-s + 9·9-s + (14.2 − 8.25i)11-s + (1.39 − 2.42i)13-s + (−10.1 − 5.88i)15-s − 2.54i·17-s − 21.5·19-s + (19.1 + 33.2i)21-s + (−2.60 − 1.50i)23-s + (−4.79 − 8.31i)25-s + 27·27-s + (−13.1 + 7.61i)29-s + (−13.6 + 23.5i)31-s + ⋯
L(s)  = 1  + 3-s + (−0.679 − 0.392i)5-s + (0.914 + 1.58i)7-s + 9-s + (1.29 − 0.750i)11-s + (0.107 − 0.186i)13-s + (−0.679 − 0.392i)15-s − 0.149i·17-s − 1.13·19-s + (0.914 + 1.58i)21-s + (−0.113 − 0.0652i)23-s + (−0.191 − 0.332i)25-s + 27-s + (−0.455 + 0.262i)29-s + (−0.438 + 0.759i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.96500 + 0.171915i\)
\(L(\frac12)\) \(\approx\) \(1.96500 + 0.171915i\)
\(L(2)\) 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 \( 1 - 3T \)
good5 \( 1 + (3.39 + 1.96i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-6.39 - 11.0i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-14.2 + 8.25i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-1.39 + 2.42i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 2.54iT - 289T^{2} \)
19 \( 1 + 21.5T + 361T^{2} \)
23 \( 1 + (2.60 + 1.50i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (13.1 - 7.61i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (13.6 - 23.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 10.4T + 1.36e3T^{2} \)
41 \( 1 + (34.5 + 19.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-17.0 - 29.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (58.1 - 33.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 100. iT - 2.80e3T^{2} \)
59 \( 1 + (5.29 + 3.05i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (20.3 + 35.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-54.4 + 94.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 52.8iT - 5.04e3T^{2} \)
73 \( 1 - 68.7T + 5.32e3T^{2} \)
79 \( 1 + (12.7 + 22.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (52.0 - 30.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 7.62iT - 7.92e3T^{2} \)
97 \( 1 + (-50.7 - 87.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.74798800946327480873103418249, −11.99230849331845252403114207908, −11.06918577329671936503220567755, −9.358406811728764692358541390362, −8.583380174774319210251472481930, −8.098417251547476018637129560909, −6.42642807949954484534789560736, −4.86997477683667786308924838923, −3.53377120617939780271939976928, −1.88074874169919798642099209355, 1.64972381781560691134461549315, 3.83211646103514031939738262936, 4.31617857100484774484524471940, 6.85414827213649745326058833936, 7.49624161054073155619483475816, 8.495040396668544743415759203984, 9.754360237611306726672230306959, 10.77814025314562062663723659373, 11.72308063893263817331264062996, 13.05485772354753564569013411623

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