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Properties

Label 2-12e2-144.59-c1-0-15
Degree $2$
Conductor $144$
Sign $0.130 + 0.991i$
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  + (0.0381 + 1.41i)2-s + (−1.59 − 0.679i)3-s + (−1.99 + 0.107i)4-s + (−0.642 − 2.39i)5-s + (0.899 − 2.27i)6-s + (−1.93 − 3.34i)7-s + (−0.228 − 2.81i)8-s + (2.07 + 2.16i)9-s + (3.36 − 0.999i)10-s + (−1.07 + 4.01i)11-s + (3.25 + 1.18i)12-s + (−0.850 − 3.17i)13-s + (4.65 − 2.85i)14-s + (−0.605 + 4.25i)15-s + (3.97 − 0.431i)16-s + 1.33i·17-s + ⋯
L(s)  = 1  + (0.0269 + 0.999i)2-s + (−0.919 − 0.392i)3-s + (−0.998 + 0.0539i)4-s + (−0.287 − 1.07i)5-s + (0.367 − 0.930i)6-s + (−0.730 − 1.26i)7-s + (−0.0809 − 0.996i)8-s + (0.692 + 0.721i)9-s + (1.06 − 0.316i)10-s + (−0.324 + 1.21i)11-s + (0.939 + 0.342i)12-s + (−0.235 − 0.880i)13-s + (1.24 − 0.764i)14-s + (−0.156 + 1.09i)15-s + (0.994 − 0.107i)16-s + 0.322i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.130 + 0.991i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.130 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.339117 - 0.297433i\)
\(L(\frac12)\) \(\approx\) \(0.339117 - 0.297433i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0381 - 1.41i)T \)
3 \( 1 + (1.59 + 0.679i)T \)
good5 \( 1 + (0.642 + 2.39i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.93 + 3.34i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.07 - 4.01i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.850 + 3.17i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 1.33iT - 17T^{2} \)
19 \( 1 + (6.09 + 6.09i)T + 19iT^{2} \)
23 \( 1 + (-0.521 - 0.301i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0730 + 0.272i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-5.84 - 3.37i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.00346 - 0.00346i)T + 37iT^{2} \)
41 \( 1 + (-0.614 + 1.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.563 - 0.151i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.24 + 2.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.24 + 3.24i)T - 53iT^{2} \)
59 \( 1 + (-5.40 + 1.44i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.97 + 0.528i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-9.90 + 2.65i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.85iT - 71T^{2} \)
73 \( 1 + 7.41iT - 73T^{2} \)
79 \( 1 + (-0.839 + 0.484i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.639 + 0.171i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (3.24 + 5.62i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.88907718712813966864609333489, −12.41816657034811505073230055574, −10.63078944895978599210780602625, −9.813712025392111253179512310611, −8.365930641107169004997211391176, −7.28901081105858364977594860245, −6.54423386204526333580698523899, −5.02596284936108854588171314247, −4.32711448840205731722768136509, −0.49870655354588888757209775808, 2.69412008080596795960356067949, 3.97336578543698135047116349476, 5.62934025295506840141376478861, 6.48878865509660466074812503460, 8.448947658022756859724505362551, 9.592598291934690770368606708790, 10.51087517060481373310910481022, 11.34231851944988317775352100965, 12.03234859489511522552574870158, 12.93529377817401434088660337230

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