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Properties

Label 2-12e2-144.61-c1-0-10
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: $\chi_{144} (61, \cdot )$
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

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

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