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Properties

Label 2-12e2-36.11-c1-0-0
Degree $2$
Conductor $144$
Sign $0.642 - 0.766i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.642 - 0.766i$
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: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16402 + 0.542795i\)
\(L(\frac12)\) \(\approx\) \(1.16402 + 0.542795i\)
\(L(\frac{3}{2})\) 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

−13.45671891087735484083981300377, −11.92664055228089504978083804730, −11.28527064559380212674489377115, −10.29710659205747403231092553134, −8.811005476416011679890369474546, −8.079960609985223141844093130107, −7.23421296916661365934298413094, −5.23466875149161686678487311624, −3.90442411212571966814144237123, −2.72349697245102963239984558617, 1.60614571663652286958938868619, 3.90799930380838217657391987606, 4.64558416188840452997018768765, 6.98773585159259695846011777266, 7.76919178146410715595661776394, 8.512409772558547881189167459734, 9.624069631987275098561113869906, 11.23722003571450827227201111174, 12.07666281357419016843448241277, 12.80998468546271284088912910412

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