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Properties

Label 2-12e2-144.85-c1-0-16
Degree $2$
Conductor $144$
Sign $0.785 + 0.619i$
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  + (1.41 + 0.0402i)2-s + (−0.162 − 1.72i)3-s + (1.99 + 0.113i)4-s + (−2.21 + 0.592i)5-s + (−0.159 − 2.44i)6-s + (2.67 − 1.54i)7-s + (2.81 + 0.241i)8-s + (−2.94 + 0.559i)9-s + (−3.14 + 0.748i)10-s + (−0.918 + 3.42i)11-s + (−0.127 − 3.46i)12-s + (−0.375 − 1.40i)13-s + (3.83 − 2.07i)14-s + (1.38 + 3.71i)15-s + (3.97 + 0.454i)16-s − 1.69·17-s + ⋯
L(s)  = 1  + (0.999 + 0.0284i)2-s + (−0.0936 − 0.995i)3-s + (0.998 + 0.0568i)4-s + (−0.988 + 0.264i)5-s + (−0.0653 − 0.997i)6-s + (1.00 − 0.582i)7-s + (0.996 + 0.0852i)8-s + (−0.982 + 0.186i)9-s + (−0.996 + 0.236i)10-s + (−0.277 + 1.03i)11-s + (−0.0368 − 0.999i)12-s + (−0.104 − 0.389i)13-s + (1.02 − 0.553i)14-s + (0.356 + 0.959i)15-s + (0.993 + 0.113i)16-s − 0.411·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.785 + 0.619i$
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: \((2,\ 144,\ (\ :1/2),\ 0.785 + 0.619i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62422 - 0.563569i\)
\(L(\frac12)\) \(\approx\) \(1.62422 - 0.563569i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0402i)T \)
3 \( 1 + (0.162 + 1.72i)T \)
good5 \( 1 + (2.21 - 0.592i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-2.67 + 1.54i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.918 - 3.42i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.375 + 1.40i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 1.69T + 17T^{2} \)
19 \( 1 + (5.41 - 5.41i)T - 19iT^{2} \)
23 \( 1 + (-3.69 - 2.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.550 + 0.147i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-3.59 + 6.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.59 - 2.59i)T + 37iT^{2} \)
41 \( 1 + (8.14 + 4.70i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.81 + 10.5i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (0.322 + 0.558i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.59 + 7.59i)T + 53iT^{2} \)
59 \( 1 + (-5.82 + 1.55i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.88 + 1.04i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.07 + 4.02i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 4.81iT - 71T^{2} \)
73 \( 1 - 0.0254iT - 73T^{2} \)
79 \( 1 + (-7.90 - 13.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.35 - 2.50i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 1.29iT - 89T^{2} \)
97 \( 1 + (-4.31 - 7.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.93453126295760576924833624989, −12.11330703633479118694219347843, −11.33896406417604130462562288356, −10.48604943306570471748191724856, −8.097836587442578301596450772857, −7.59394342122040067862276625267, −6.60459701883871086057374754109, −5.09254542107254150146332036287, −3.86510203634824760660736628098, −2.02118744821353972885242196855, 2.88379840779246444367737593130, 4.37696384511606196141792540417, 4.97907853104422115691797552920, 6.38953851742285131498924319117, 8.079260011262745152662514599041, 8.871485709426963153088486682179, 10.74342577482881848900980861295, 11.26713758678768697576573175329, 11.97583342774489114928872572645, 13.24341021794894350853517197266

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