# Properties

 Degree $2$ Conductor $144$ Sign $0.243 - 0.969i$ Motivic weight $2$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−1.87 − 0.697i)2-s + (3.02 + 2.61i)4-s + (5.24 + 5.24i)5-s − 5.32·7-s + (−3.85 − 7.01i)8-s + (−6.17 − 13.4i)10-s + (−12.2 + 12.2i)11-s + (−5.73 + 5.73i)13-s + (9.98 + 3.71i)14-s + (2.33 + 15.8i)16-s + 23.3·17-s + (11.7 + 11.7i)19-s + (2.17 + 29.5i)20-s + (31.5 − 14.4i)22-s − 5.80·23-s + ⋯
 L(s)  = 1 + (−0.937 − 0.348i)2-s + (0.757 + 0.653i)4-s + (1.04 + 1.04i)5-s − 0.761·7-s + (−0.481 − 0.876i)8-s + (−0.617 − 1.34i)10-s + (−1.11 + 1.11i)11-s + (−0.441 + 0.441i)13-s + (0.713 + 0.265i)14-s + (0.146 + 0.989i)16-s + 1.37·17-s + (0.618 + 0.618i)19-s + (0.108 + 1.47i)20-s + (1.43 − 0.657i)22-s − 0.252·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$144$$    =    $$2^{4} \cdot 3^{2}$$ Sign: $0.243 - 0.969i$ Motivic weight: $$2$$ Character: $\chi_{144} (91, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 144,\ (\ :1),\ 0.243 - 0.969i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.694100 + 0.541393i$$ $$L(\frac12)$$ $$\approx$$ $$0.694100 + 0.541393i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.87 + 0.697i)T$$
3 $$1$$
good5 $$1 + (-5.24 - 5.24i)T + 25iT^{2}$$
7 $$1 + 5.32T + 49T^{2}$$
11 $$1 + (12.2 - 12.2i)T - 121iT^{2}$$
13 $$1 + (5.73 - 5.73i)T - 169iT^{2}$$
17 $$1 - 23.3T + 289T^{2}$$
19 $$1 + (-11.7 - 11.7i)T + 361iT^{2}$$
23 $$1 + 5.80T + 529T^{2}$$
29 $$1 + (18.3 - 18.3i)T - 841iT^{2}$$
31 $$1 - 16.9iT - 961T^{2}$$
37 $$1 + (-15.3 - 15.3i)T + 1.36e3iT^{2}$$
41 $$1 + 29.2iT - 1.68e3T^{2}$$
43 $$1 + (-33.4 + 33.4i)T - 1.84e3iT^{2}$$
47 $$1 + 18.2iT - 2.20e3T^{2}$$
53 $$1 + (-66.9 - 66.9i)T + 2.80e3iT^{2}$$
59 $$1 + (-27.1 + 27.1i)T - 3.48e3iT^{2}$$
61 $$1 + (-65.2 + 65.2i)T - 3.72e3iT^{2}$$
67 $$1 + (37.6 + 37.6i)T + 4.48e3iT^{2}$$
71 $$1 + 42.6T + 5.04e3T^{2}$$
73 $$1 + 106. iT - 5.32e3T^{2}$$
79 $$1 - 21.2iT - 6.24e3T^{2}$$
83 $$1 + (24.1 + 24.1i)T + 6.88e3iT^{2}$$
89 $$1 - 52.8iT - 7.92e3T^{2}$$
97 $$1 + 21.0T + 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

−12.84174693868091380389140516443, −12.03128934413930111734748420054, −10.51932975942703913554360819833, −10.07532737191160124113194384133, −9.376280071058669097279653715284, −7.65970668680825106195182778336, −6.91412534827963263430886338535, −5.65365524298172145440208653984, −3.23829373841088735411106114940, −2.08722869571862524433982341014, 0.76297326103762787394005385553, 2.73000152537449838374614107167, 5.41237667505948924686997216195, 5.89336514902468575432167746147, 7.55481144706274935005954415816, 8.544075682570909977111968107076, 9.647936922588478836167344664267, 10.09367486673953932345483122815, 11.46468951604371630613360675614, 12.82139054159663593407593257750