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Properties

Label 2-12e2-48.5-c2-0-8
Degree $2$
Conductor $144$
Sign $-0.161 + 0.986i$
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  + (−1.76 − 0.939i)2-s + (2.23 + 3.31i)4-s + (−6.08 + 6.08i)5-s − 9.40i·7-s + (−0.828 − 7.95i)8-s + (16.4 − 5.02i)10-s + (11.9 − 11.9i)11-s + (3.47 − 3.47i)13-s + (−8.83 + 16.5i)14-s + (−6.01 + 14.8i)16-s − 28.5i·17-s + (−3.08 + 3.08i)19-s + (−33.7 − 6.59i)20-s + (−32.3 + 9.87i)22-s + 2.57·23-s + ⋯
L(s)  = 1  + (−0.882 − 0.469i)2-s + (0.558 + 0.829i)4-s + (−1.21 + 1.21i)5-s − 1.34i·7-s + (−0.103 − 0.994i)8-s + (1.64 − 0.502i)10-s + (1.08 − 1.08i)11-s + (0.267 − 0.267i)13-s + (−0.630 + 1.18i)14-s + (−0.375 + 0.926i)16-s − 1.67i·17-s + (−0.162 + 0.162i)19-s + (−1.68 − 0.329i)20-s + (−1.46 + 0.448i)22-s + 0.111·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.161 + 0.986i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.161 + 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.414484 - 0.487957i\)
\(L(\frac12)\) \(\approx\) \(0.414484 - 0.487957i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.76 + 0.939i)T \)
3 \( 1 \)
good5 \( 1 + (6.08 - 6.08i)T - 25iT^{2} \)
7 \( 1 + 9.40iT - 49T^{2} \)
11 \( 1 + (-11.9 + 11.9i)T - 121iT^{2} \)
13 \( 1 + (-3.47 + 3.47i)T - 169iT^{2} \)
17 \( 1 + 28.5iT - 289T^{2} \)
19 \( 1 + (3.08 - 3.08i)T - 361iT^{2} \)
23 \( 1 - 2.57T + 529T^{2} \)
29 \( 1 + (-3.49 - 3.49i)T + 841iT^{2} \)
31 \( 1 + 21.0T + 961T^{2} \)
37 \( 1 + (13.2 + 13.2i)T + 1.36e3iT^{2} \)
41 \( 1 + 11.2T + 1.68e3T^{2} \)
43 \( 1 + (-8.19 - 8.19i)T + 1.84e3iT^{2} \)
47 \( 1 + 17.2iT - 2.20e3T^{2} \)
53 \( 1 + (30.5 - 30.5i)T - 2.80e3iT^{2} \)
59 \( 1 + (-14.7 + 14.7i)T - 3.48e3iT^{2} \)
61 \( 1 + (-39.3 + 39.3i)T - 3.72e3iT^{2} \)
67 \( 1 + (-79.6 + 79.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 73.1T + 5.04e3T^{2} \)
73 \( 1 - 1.23iT - 5.32e3T^{2} \)
79 \( 1 + 110.T + 6.24e3T^{2} \)
83 \( 1 + (13.7 + 13.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 39.2T + 7.92e3T^{2} \)
97 \( 1 + 109.T + 9.40e3T^{2} \)
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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

−12.11232966963366398530482353359, −11.12866810506197149559732137122, −10.91711093771720394298884623441, −9.625533931982080384749293999731, −8.287149705183004676611795189030, −7.29325893006287512695817177029, −6.69804808109924546374245936393, −3.93787656612160372946880891719, −3.18930069178026312276195062069, −0.58935934977416092107028793753, 1.63987580380035813500105055863, 4.20944564085662938556131839061, 5.55372082460436508869124387057, 6.88179911489570967666214763325, 8.270930109568016524002202456062, 8.731377774246890145054643393684, 9.651461273133814188371855346974, 11.24449125581499541541326895636, 12.08614221157851163499358595112, 12.70596549499898495761265970144

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