Properties

Degree $2$
Conductor $144$
Sign $0.999 + 0.0338i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.25 − 1.55i)2-s + (−0.846 + 3.90i)4-s + (−0.909 + 0.909i)5-s − 0.654·7-s + (7.14 − 3.59i)8-s + (2.55 + 0.273i)10-s + (13.3 + 13.3i)11-s + (8.32 + 8.32i)13-s + (0.822 + 1.01i)14-s + (−14.5 − 6.62i)16-s + 3.93·17-s + (16.8 − 16.8i)19-s + (−2.78 − 4.32i)20-s + (4.02 − 37.6i)22-s + 23.1·23-s + ⋯
L(s)  = 1  + (−0.627 − 0.778i)2-s + (−0.211 + 0.977i)4-s + (−0.181 + 0.181i)5-s − 0.0935·7-s + (0.893 − 0.448i)8-s + (0.255 + 0.0273i)10-s + (1.21 + 1.21i)11-s + (0.640 + 0.640i)13-s + (0.0587 + 0.0728i)14-s + (−0.910 − 0.413i)16-s + 0.231·17-s + (0.889 − 0.889i)19-s + (−0.139 − 0.216i)20-s + (0.183 − 1.70i)22-s + 1.00·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0338i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.999 + 0.0338i$
Motivic weight: \(2\)
Character: $\chi_{144} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.999 + 0.0338i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.04825 - 0.0177720i\)
\(L(\frac12)\) \(\approx\) \(1.04825 - 0.0177720i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 + 1.55i)T \)
3 \( 1 \)
good5 \( 1 + (0.909 - 0.909i)T - 25iT^{2} \)
7 \( 1 + 0.654T + 49T^{2} \)
11 \( 1 + (-13.3 - 13.3i)T + 121iT^{2} \)
13 \( 1 + (-8.32 - 8.32i)T + 169iT^{2} \)
17 \( 1 - 3.93T + 289T^{2} \)
19 \( 1 + (-16.8 + 16.8i)T - 361iT^{2} \)
23 \( 1 - 23.1T + 529T^{2} \)
29 \( 1 + (35.6 + 35.6i)T + 841iT^{2} \)
31 \( 1 - 45.5iT - 961T^{2} \)
37 \( 1 + (-10.1 + 10.1i)T - 1.36e3iT^{2} \)
41 \( 1 - 28.4iT - 1.68e3T^{2} \)
43 \( 1 + (-22.7 - 22.7i)T + 1.84e3iT^{2} \)
47 \( 1 + 10.7iT - 2.20e3T^{2} \)
53 \( 1 + (41.5 - 41.5i)T - 2.80e3iT^{2} \)
59 \( 1 + (-21.0 - 21.0i)T + 3.48e3iT^{2} \)
61 \( 1 + (68.7 + 68.7i)T + 3.72e3iT^{2} \)
67 \( 1 + (-67.8 + 67.8i)T - 4.48e3iT^{2} \)
71 \( 1 + 33.3T + 5.04e3T^{2} \)
73 \( 1 + 18.6iT - 5.32e3T^{2} \)
79 \( 1 - 6.29iT - 6.24e3T^{2} \)
83 \( 1 + (-72.0 + 72.0i)T - 6.88e3iT^{2} \)
89 \( 1 + 10.6iT - 7.92e3T^{2} \)
97 \( 1 - 143.T + 9.40e3T^{2} \)
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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

−12.62024312973568063759638026348, −11.63611851681291137053270565685, −11.00654174896036306220879644417, −9.526985909318899981550404798098, −9.162853366591946773079593969238, −7.60603635403062537086237685306, −6.71997090172633183415682935251, −4.61180020966965048895209721159, −3.34214014808263500421882144760, −1.52385376132949931977552105724, 1.02146925367122762901840987462, 3.68938642318005316063893050996, 5.46684133858223835827953500284, 6.36660832090504608841269492824, 7.67907791018642994690443039891, 8.649156996395107440752134323700, 9.487666916040145870658749463190, 10.74503668186257016354308860315, 11.62493248415321370944165296595, 13.07609057923098832561431543482

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