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Properties

Label 2-12e2-144.5-c2-0-11
Degree $2$
Conductor $144$
Sign $0.814 - 0.580i$
Analytic cond. $3.92371$
Root an. cond. $1.98083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.09 − 1.67i)2-s + (0.894 + 2.86i)3-s + (−1.58 + 3.67i)4-s + (−0.835 − 3.11i)5-s + (3.80 − 4.64i)6-s + (0.108 − 0.0629i)7-s + (7.87 − 1.38i)8-s + (−7.40 + 5.12i)9-s + (−4.29 + 4.82i)10-s + (16.3 + 4.38i)11-s + (−11.9 − 1.25i)12-s + (6.21 + 23.2i)13-s + (−0.224 − 0.112i)14-s + (8.18 − 5.18i)15-s + (−10.9 − 11.6i)16-s + 26.4i·17-s + ⋯
L(s)  = 1  + (−0.549 − 0.835i)2-s + (0.298 + 0.954i)3-s + (−0.396 + 0.918i)4-s + (−0.167 − 0.623i)5-s + (0.633 − 0.773i)6-s + (0.0155 − 0.00898i)7-s + (0.984 − 0.173i)8-s + (−0.822 + 0.568i)9-s + (−0.429 + 0.482i)10-s + (1.48 + 0.398i)11-s + (−0.994 − 0.104i)12-s + (0.478 + 1.78i)13-s + (−0.0160 − 0.00807i)14-s + (0.545 − 0.345i)15-s + (−0.685 − 0.727i)16-s + 1.55i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.814 - 0.580i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.814 - 0.580i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.06630 + 0.340940i\)
\(L(\frac12)\) \(\approx\) \(1.06630 + 0.340940i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.09 + 1.67i)T \)
3 \( 1 + (-0.894 - 2.86i)T \)
good5 \( 1 + (0.835 + 3.11i)T + (-21.6 + 12.5i)T^{2} \)
7 \( 1 + (-0.108 + 0.0629i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-16.3 - 4.38i)T + (104. + 60.5i)T^{2} \)
13 \( 1 + (-6.21 - 23.2i)T + (-146. + 84.5i)T^{2} \)
17 \( 1 - 26.4iT - 289T^{2} \)
19 \( 1 + (12.4 - 12.4i)T - 361iT^{2} \)
23 \( 1 + (-11.0 + 19.0i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-9.19 + 34.3i)T + (-728. - 420.5i)T^{2} \)
31 \( 1 + (2.65 - 4.59i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (22.6 + 22.6i)T + 1.36e3iT^{2} \)
41 \( 1 + (13.0 - 22.5i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-5.71 + 21.3i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-13.6 + 7.86i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-19.5 + 19.5i)T - 2.80e3iT^{2} \)
59 \( 1 + (14.4 + 53.8i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-74.0 - 19.8i)T + (3.22e3 + 1.86e3i)T^{2} \)
67 \( 1 + (14.5 + 54.1i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 10.7T + 5.04e3T^{2} \)
73 \( 1 - 98.0iT - 5.32e3T^{2} \)
79 \( 1 + (-1.02 - 1.77i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-23.1 + 86.3i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 + 76.9T + 7.92e3T^{2} \)
97 \( 1 + (-19.6 - 34.0i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.64380626310274616072986065024, −11.78994101108218840632403705695, −10.86987320060257784049188109637, −9.809478470878333165969212882754, −8.878940079898156267747264989775, −8.416846222990833014629572956709, −6.52877112015634274885829810149, −4.37064670628122168859316740788, −3.93528448586254629154448148478, −1.80511405934044078800448537181, 0.953336492801971182425643645204, 3.18140227701504226999704801368, 5.41269869670298118047469669598, 6.65942723896701409709901673277, 7.26257612799615147661365947436, 8.464951159933842733262573778664, 9.254239386776144812677954472485, 10.71884163394244177347435976147, 11.64724555529297800283220860573, 13.06806157201045039694105207567

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