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Properties

Label 2-12e2-9.5-c2-0-5
Degree $2$
Conductor $144$
Sign $0.959 + 0.283i$
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  + (0.686 − 2.92i)3-s + (6.55 + 3.78i)5-s + (4.55 + 7.89i)7-s + (−8.05 − 4.00i)9-s + (0.383 − 0.221i)11-s + (5.55 − 9.62i)13-s + (15.5 − 16.5i)15-s − 8.01i·17-s + 8.11·19-s + (26.1 − 7.89i)21-s + (−20.4 − 11.8i)23-s + (16.1 + 28.0i)25-s + (−17.2 + 20.7i)27-s + (−45.9 + 26.5i)29-s + (14.6 − 25.4i)31-s + ⋯
L(s)  = 1  + (0.228 − 0.973i)3-s + (1.31 + 0.757i)5-s + (0.651 + 1.12i)7-s + (−0.895 − 0.445i)9-s + (0.0348 − 0.0201i)11-s + (0.427 − 0.740i)13-s + (1.03 − 1.10i)15-s − 0.471i·17-s + 0.427·19-s + (1.24 − 0.375i)21-s + (−0.888 − 0.513i)23-s + (0.647 + 1.12i)25-s + (−0.638 + 0.769i)27-s + (−1.58 + 0.913i)29-s + (0.473 − 0.819i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.959 + 0.283i$
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: \((2,\ 144,\ (\ :1),\ 0.959 + 0.283i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.87709 - 0.271207i\)
\(L(\frac12)\) \(\approx\) \(1.87709 - 0.271207i\)
\(L(2)\) 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 \( 1 + (-0.686 + 2.92i)T \)
good5 \( 1 + (-6.55 - 3.78i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-4.55 - 7.89i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-0.383 + 0.221i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-5.55 + 9.62i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 8.01iT - 289T^{2} \)
19 \( 1 - 8.11T + 361T^{2} \)
23 \( 1 + (20.4 + 11.8i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (45.9 - 26.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-14.6 + 25.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 18.4T + 1.36e3T^{2} \)
41 \( 1 + (38.9 + 22.4i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-11.5 - 19.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-7.32 + 4.22i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 60.5iT - 2.80e3T^{2} \)
59 \( 1 + (65.9 + 38.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (2.67 + 4.63i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (54.8 - 95.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 16.0iT - 5.04e3T^{2} \)
73 \( 1 + 4.35T + 5.32e3T^{2} \)
79 \( 1 + (-0.792 - 1.37i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-7.32 + 4.22i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 64.1iT - 7.92e3T^{2} \)
97 \( 1 + (57.6 + 99.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.93969745088065598722857631566, −11.89664556173828628074230820984, −10.90150357759917269993579709183, −9.606747661679046421370270945169, −8.611465574103012347048495433090, −7.45879734614218366091815576183, −6.13356583486110811311867919421, −5.53074998367940250915103558501, −2.87458207401416347660618943505, −1.83728588167305081205276489743, 1.72057578928932243733779162326, 3.89939669919011737469950814302, 4.96542777534912950258757248697, 6.09318878281470236542862062003, 7.83874043363909364404563112541, 9.011834624299118780574931708676, 9.806351726874092048210547056385, 10.63655144334983341963746288698, 11.70760362829778180680747860314, 13.35997180460860879922012340907

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