Properties

Label 2-12e2-144.85-c1-0-3
Degree $2$
Conductor $144$
Sign $-0.364 - 0.931i$
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  + (1.11 + 0.868i)2-s + (−0.795 + 1.53i)3-s + (0.492 + 1.93i)4-s + (−2.41 + 0.646i)5-s + (−2.22 + 1.02i)6-s + (2.82 − 1.62i)7-s + (−1.13 + 2.59i)8-s + (−1.73 − 2.44i)9-s + (−3.25 − 1.37i)10-s + (0.356 − 1.32i)11-s + (−3.37 − 0.783i)12-s + (1.42 + 5.32i)13-s + (4.56 + 0.630i)14-s + (0.924 − 4.22i)15-s + (−3.51 + 1.91i)16-s + 5.37·17-s + ⋯
L(s)  = 1  + (0.789 + 0.613i)2-s + (−0.459 + 0.888i)3-s + (0.246 + 0.969i)4-s + (−1.07 + 0.289i)5-s + (−0.907 + 0.419i)6-s + (1.06 − 0.615i)7-s + (−0.400 + 0.916i)8-s + (−0.578 − 0.815i)9-s + (−1.02 − 0.434i)10-s + (0.107 − 0.400i)11-s + (−0.974 − 0.226i)12-s + (0.396 + 1.47i)13-s + (1.22 + 0.168i)14-s + (0.238 − 1.09i)15-s + (−0.878 + 0.477i)16-s + 1.30·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.364 - 0.931i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.364 - 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.754443 + 1.10524i\)
\(L(\frac12)\) \(\approx\) \(0.754443 + 1.10524i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.11 - 0.868i)T \)
3 \( 1 + (0.795 - 1.53i)T \)
good5 \( 1 + (2.41 - 0.646i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-2.82 + 1.62i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.356 + 1.32i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.42 - 5.32i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 5.37T + 17T^{2} \)
19 \( 1 + (-4.71 + 4.71i)T - 19iT^{2} \)
23 \( 1 + (2.88 + 1.66i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.03 + 0.814i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-0.621 + 1.07i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.86 + 5.86i)T + 37iT^{2} \)
41 \( 1 + (2.81 + 1.62i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.61 + 6.03i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-2.17 - 3.76i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.134 + 0.134i)T + 53iT^{2} \)
59 \( 1 + (2.21 - 0.592i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.29 + 0.615i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.0300 + 0.112i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.21iT - 71T^{2} \)
73 \( 1 - 9.75iT - 73T^{2} \)
79 \( 1 + (-1.11 - 1.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.74 - 1.54i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 6.12iT - 89T^{2} \)
97 \( 1 + (2.21 + 3.83i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.92550693606676533043134358782, −12.03797858355511157925627337445, −11.55271141539176289320864702043, −10.83044329156474713811286265826, −9.110461778978960739517546381852, −7.907037398481279490853697393911, −6.96674607957217473824024358798, −5.47850536492007714634706505681, −4.34290549746484289352107541506, −3.58711421976688960718084464846, 1.40574900883502613608654854743, 3.36772923371632772323469318794, 5.06611823439960373126894274854, 5.79669493866983394063078426963, 7.58712430709463146370182971053, 8.203789216819802333208512537479, 10.12783414386255531550791729473, 11.26219392592479382166527561682, 12.11650771305013634329045899388, 12.29789115018299170447205538282

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