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Properties

Label 2-12e2-48.35-c3-0-9
Degree $2$
Conductor $144$
Sign $0.571 - 0.820i$
Analytic cond. $8.49627$
Root an. cond. $2.91483$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.272 + 2.81i)2-s + (−7.85 − 1.53i)4-s + (−6.30 − 6.30i)5-s + 27.2·7-s + (6.45 − 21.6i)8-s + (19.4 − 16.0i)10-s + (4.03 − 4.03i)11-s + (37.9 + 37.9i)13-s + (−7.41 + 76.5i)14-s + (59.2 + 24.0i)16-s + 79.8i·17-s + (75.2 − 75.2i)19-s + (39.8 + 59.2i)20-s + (10.2 + 12.4i)22-s + 25.2i·23-s + ⋯
L(s)  = 1  + (−0.0963 + 0.995i)2-s + (−0.981 − 0.191i)4-s + (−0.564 − 0.564i)5-s + 1.46·7-s + (0.285 − 0.958i)8-s + (0.616 − 0.507i)10-s + (0.110 − 0.110i)11-s + (0.809 + 0.809i)13-s + (−0.141 + 1.46i)14-s + (0.926 + 0.376i)16-s + 1.13i·17-s + (0.908 − 0.908i)19-s + (0.445 + 0.662i)20-s + (0.0995 + 0.120i)22-s + 0.228i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.571 - 0.820i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.571 - 0.820i$
Analytic conductor: \(8.49627\)
Root analytic conductor: \(2.91483\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :3/2),\ 0.571 - 0.820i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.38194 + 0.721227i\)
\(L(\frac12)\) \(\approx\) \(1.38194 + 0.721227i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.272 - 2.81i)T \)
3 \( 1 \)
good5 \( 1 + (6.30 + 6.30i)T + 125iT^{2} \)
7 \( 1 - 27.2T + 343T^{2} \)
11 \( 1 + (-4.03 + 4.03i)T - 1.33e3iT^{2} \)
13 \( 1 + (-37.9 - 37.9i)T + 2.19e3iT^{2} \)
17 \( 1 - 79.8iT - 4.91e3T^{2} \)
19 \( 1 + (-75.2 + 75.2i)T - 6.85e3iT^{2} \)
23 \( 1 - 25.2iT - 1.21e4T^{2} \)
29 \( 1 + (-107. + 107. i)T - 2.43e4iT^{2} \)
31 \( 1 - 237. iT - 2.97e4T^{2} \)
37 \( 1 + (-210. + 210. i)T - 5.06e4iT^{2} \)
41 \( 1 - 378.T + 6.89e4T^{2} \)
43 \( 1 + (-191. - 191. i)T + 7.95e4iT^{2} \)
47 \( 1 + 417.T + 1.03e5T^{2} \)
53 \( 1 + (139. + 139. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-282. + 282. i)T - 2.05e5iT^{2} \)
61 \( 1 + (255. + 255. i)T + 2.26e5iT^{2} \)
67 \( 1 + (348. - 348. i)T - 3.00e5iT^{2} \)
71 \( 1 + 321. iT - 3.57e5T^{2} \)
73 \( 1 - 135. iT - 3.89e5T^{2} \)
79 \( 1 + 522. iT - 4.93e5T^{2} \)
83 \( 1 + (444. + 444. i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.10e3T + 7.04e5T^{2} \)
97 \( 1 + 1.06e3T + 9.12e5T^{2} \)
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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

−12.97265822467497466652651157155, −11.76698869781438619170368397466, −10.82656321469555458207310957079, −9.235028857015121986680084936756, −8.374515737559560945073793062284, −7.69538414562046564661126613759, −6.29198157599410656966974126251, −4.94577970586384372463256442012, −4.10733516457077005662489092284, −1.18179414511522910848863702991, 1.16051551942493853198171015163, 2.91414170566925825727075799599, 4.26123228313758968707020688528, 5.49107844023758231942379392098, 7.57676672421918965422177030228, 8.234053878536218928055698673785, 9.558247878940144834214447560228, 10.78940022787935795058995716051, 11.36201038791652139618541957620, 12.11872645881127442209703529245

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