# Properties

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

# Related objects

## 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