# Properties

 Label 2-38e2-19.2-c0-0-0 Degree $2$ Conductor $1444$ Sign $0.950 + 0.309i$ Analytic cond. $0.720649$ Root an. cond. $0.848910$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.909 + 1.08i)3-s + (−0.939 + 0.342i)5-s + (−0.5 − 0.866i)7-s + (−0.173 − 0.984i)9-s + (0.5 − 0.866i)11-s + (0.483 − 1.32i)15-s + (0.173 − 0.984i)17-s + (1.39 + 0.245i)21-s + (1.39 − 0.245i)29-s + (0.483 + 1.32i)33-s + (0.766 + 0.642i)35-s + 1.41i·37-s + (0.909 − 1.08i)41-s + (0.939 − 0.342i)43-s + (0.499 + 0.866i)45-s + ⋯
 L(s)  = 1 + (−0.909 + 1.08i)3-s + (−0.939 + 0.342i)5-s + (−0.5 − 0.866i)7-s + (−0.173 − 0.984i)9-s + (0.5 − 0.866i)11-s + (0.483 − 1.32i)15-s + (0.173 − 0.984i)17-s + (1.39 + 0.245i)21-s + (1.39 − 0.245i)29-s + (0.483 + 1.32i)33-s + (0.766 + 0.642i)35-s + 1.41i·37-s + (0.909 − 1.08i)41-s + (0.939 − 0.342i)43-s + (0.499 + 0.866i)45-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1444$$    =    $$2^{2} \cdot 19^{2}$$ Sign: $0.950 + 0.309i$ Analytic conductor: $$0.720649$$ Root analytic conductor: $$0.848910$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1444} (477, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1444,\ (\ :0),\ 0.950 + 0.309i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5527035481$$ $$L(\frac12)$$ $$\approx$$ $$0.5527035481$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
19 $$1$$
good3 $$1 + (0.909 - 1.08i)T + (-0.173 - 0.984i)T^{2}$$
5 $$1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2}$$
7 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
11 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
13 $$1 + (-0.173 + 0.984i)T^{2}$$
17 $$1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2}$$
23 $$1 + (0.766 + 0.642i)T^{2}$$
29 $$1 + (-1.39 + 0.245i)T + (0.939 - 0.342i)T^{2}$$
31 $$1 + (0.5 - 0.866i)T^{2}$$
37 $$1 - 1.41iT - T^{2}$$
41 $$1 + (-0.909 + 1.08i)T + (-0.173 - 0.984i)T^{2}$$
43 $$1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2}$$
47 $$1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2}$$
53 $$1 + (-0.766 - 0.642i)T^{2}$$
59 $$1 + (1.39 + 0.245i)T + (0.939 + 0.342i)T^{2}$$
61 $$1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2}$$
67 $$1 + (0.939 - 0.342i)T^{2}$$
71 $$1 + (0.483 + 1.32i)T + (-0.766 + 0.642i)T^{2}$$
73 $$1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2}$$
79 $$1 + (-0.173 - 0.984i)T^{2}$$
83 $$1 + (-0.5 + 0.866i)T^{2}$$
89 $$1 + (-0.173 + 0.984i)T^{2}$$
97 $$1 + (0.939 + 0.342i)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

−9.907986420764233491246993007909, −9.065019587788052415263148853992, −8.040895052874293892488232812874, −7.14010723373297638118689353836, −6.40200757507579820510886989513, −5.44872590080901432395967857384, −4.47764294243128945348665231339, −3.84208467239671716717627394613, −3.05602487638446396071454604385, −0.60234090160508946529768394078, 1.18872135141460663158234252029, 2.47239845984761214581065418652, 3.88744766739949347227775719979, 4.80358021270631337859934608562, 5.96372369061554460994726869877, 6.36385363333305876303252482760, 7.34973757676725664543100045242, 7.945879951877690470074969176289, 8.871430546550454667649301731901, 9.685345587813347444910709860008