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Properties

Label 2-12e2-36.11-c1-0-5
Degree $2$
Conductor $144$
Sign $-0.984 - 0.173i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)3-s + (−3 + 1.73i)5-s + (−3 − 1.73i)7-s + (1.5 + 2.59i)9-s + (−1.5 + 2.59i)11-s + (−2 − 3.46i)13-s + 6·15-s − 1.73i·17-s + 1.73i·19-s + (3 + 5.19i)21-s + (3.5 − 6.06i)25-s − 5.19i·27-s + (−3 − 1.73i)29-s + (4.5 − 2.59i)33-s + 12·35-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s + (−1.34 + 0.774i)5-s + (−1.13 − 0.654i)7-s + (0.5 + 0.866i)9-s + (−0.452 + 0.783i)11-s + (−0.554 − 0.960i)13-s + 1.54·15-s − 0.420i·17-s + 0.397i·19-s + (0.654 + 1.13i)21-s + (0.700 − 1.21i)25-s − 0.999i·27-s + (−0.557 − 0.321i)29-s + (0.783 − 0.452i)33-s + 2.02·35-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.984 - 0.173i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 0.866i)T \)
good5 \( 1 + (3 - 1.73i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (3 + 1.73i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.73iT - 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 1.73i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.5 - 2.59i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6 - 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (7.5 + 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.5 - 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + (-3 - 1.73i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.8iT - 89T^{2} \)
97 \( 1 + (6.5 - 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.53738723894124325322831459464, −11.61037786181095395031438337570, −10.62173271796120513717245260138, −9.882101617114231288055950086789, −7.71317973727378454066988344492, −7.34671063527856289449570099151, −6.21303297723190117732479373272, −4.57699756012352974507332297910, −3.10905564533405997568619502780, 0, 3.48837582993728922263359163609, 4.67338578996700525503174029078, 5.89872616341568826783990414528, 7.14890516216849682363567770269, 8.631598278225329330423059223842, 9.455304537187462249333404721519, 10.76713271626389253569414382272, 11.81104424158415606724684820311, 12.30401005689130514723881916050

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