# Properties

 Label 2-1638-7.4-c1-0-28 Degree $2$ Conductor $1638$ Sign $0.605 + 0.795i$ Analytic cond. $13.0794$ Root an. cond. $3.61655$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−2.5 − 0.866i)7-s − 0.999·8-s + (−2.5 + 4.33i)11-s − 13-s + (−0.500 − 2.59i)14-s + (−0.5 − 0.866i)16-s + (3.5 − 6.06i)17-s + (−3.5 − 6.06i)19-s − 5·22-s + (1 + 1.73i)23-s + (2.5 − 4.33i)25-s + (−0.5 − 0.866i)26-s + (1.99 − 1.73i)28-s + 9·29-s + ⋯
 L(s)  = 1 + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.944 − 0.327i)7-s − 0.353·8-s + (−0.753 + 1.30i)11-s − 0.277·13-s + (−0.133 − 0.694i)14-s + (−0.125 − 0.216i)16-s + (0.848 − 1.47i)17-s + (−0.802 − 1.39i)19-s − 1.06·22-s + (0.208 + 0.361i)23-s + (0.5 − 0.866i)25-s + (−0.0980 − 0.169i)26-s + (0.377 − 0.327i)28-s + 1.67·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1638$$    =    $$2 \cdot 3^{2} \cdot 7 \cdot 13$$ Sign: $0.605 + 0.795i$ Analytic conductor: $$13.0794$$ Root analytic conductor: $$3.61655$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1638} (235, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1638,\ (\ :1/2),\ 0.605 + 0.795i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.020451030$$ $$L(\frac12)$$ $$\approx$$ $$1.020451030$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.5 - 0.866i)T$$
3 $$1$$
7 $$1 + (2.5 + 0.866i)T$$
13 $$1 + T$$
good5 $$1 + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2}$$
17 $$1 + (-3.5 + 6.06i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 - 9T + 29T^{2}$$
31 $$1 + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + 4T + 41T^{2}$$
43 $$1 - 2T + 43T^{2}$$
47 $$1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-0.5 + 0.866i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 9T + 71T^{2}$$
73 $$1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + 16T + 83T^{2}$$
89 $$1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 - 6T + 97T^{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

−9.295491744358619932618581458772, −8.387192099684624494284264448779, −7.25946836630580690033140738911, −7.07025445252110664646804122261, −6.14654595655446289822978818093, −4.90140638607968982471553293470, −4.64080503531638618810366248364, −3.21109064944812983287736315743, −2.47461252438090593952243785767, −0.36097349834419984743586074329, 1.29561055762111638574518192723, 2.75902094013391765351294794265, 3.34940118302461705559251281734, 4.30431630219931507322813451246, 5.64903030720755033003214975341, 5.91383208341202130680338410728, 6.92656342783635747472832860635, 8.362730255356323489181195412855, 8.481213636942509685412672167072, 9.764717050405730391949446399441