# Properties

 Label 2-819-91.4-c1-0-31 Degree $2$ Conductor $819$ Sign $0.639 + 0.768i$ Analytic cond. $6.53974$ Root an. cond. $2.55729$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 0.499i·2-s + 1.75·4-s + (0.902 − 0.521i)5-s + (2.63 − 0.239i)7-s − 1.87i·8-s + (−0.260 − 0.451i)10-s + (3.43 − 1.98i)11-s + (−3.57 + 0.468i)13-s + (−0.119 − 1.31i)14-s + 2.56·16-s − 0.142·17-s + (−4.77 − 2.75i)19-s + (1.57 − 0.912i)20-s + (−0.991 − 1.71i)22-s + 4.39·23-s + ⋯
 L(s)  = 1 − 0.353i·2-s + 0.875·4-s + (0.403 − 0.233i)5-s + (0.995 − 0.0904i)7-s − 0.662i·8-s + (−0.0824 − 0.142i)10-s + (1.03 − 0.598i)11-s + (−0.991 + 0.129i)13-s + (−0.0319 − 0.352i)14-s + 0.640·16-s − 0.0344·17-s + (−1.09 − 0.632i)19-s + (0.353 − 0.203i)20-s + (−0.211 − 0.366i)22-s + 0.915·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 + 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$819$$    =    $$3^{2} \cdot 7 \cdot 13$$ Sign: $0.639 + 0.768i$ Analytic conductor: $$6.53974$$ Root analytic conductor: $$2.55729$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{819} (550, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 819,\ (\ :1/2),\ 0.639 + 0.768i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.07142 - 0.971374i$$ $$L(\frac12)$$ $$\approx$$ $$2.07142 - 0.971374i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1 + (-2.63 + 0.239i)T$$
13 $$1 + (3.57 - 0.468i)T$$
good2 $$1 + 0.499iT - 2T^{2}$$
5 $$1 + (-0.902 + 0.521i)T + (2.5 - 4.33i)T^{2}$$
11 $$1 + (-3.43 + 1.98i)T + (5.5 - 9.52i)T^{2}$$
17 $$1 + 0.142T + 17T^{2}$$
19 $$1 + (4.77 + 2.75i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 - 4.39T + 23T^{2}$$
29 $$1 + (4.19 - 7.27i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (-2.46 - 1.42i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 + 0.843iT - 37T^{2}$$
41 $$1 + (10.4 + 6.04i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (-2.41 - 4.17i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (-3.94 + 2.27i)T + (23.5 - 40.7i)T^{2}$$
53 $$1 + (0.139 - 0.242i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 - 10.7iT - 59T^{2}$$
61 $$1 + (-2.93 + 5.07i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (4.45 - 2.57i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + (-3.20 + 1.84i)T + (35.5 - 61.4i)T^{2}$$
73 $$1 + (-5.72 - 3.30i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (5.96 + 10.3i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 2.87iT - 83T^{2}$$
89 $$1 + 1.74iT - 89T^{2}$$
97 $$1 + (-2.34 + 1.35i)T + (48.5 - 84.0i)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

−10.33679718251333668209165829718, −9.226301718589092606019844242215, −8.598521807265530850298809199720, −7.34718722154842585625913104751, −6.80312045903546418282158465199, −5.68577173354688012136139347483, −4.73669179561711704648903238483, −3.52923126661019591506768901975, −2.25482742446617083838257937533, −1.30725486933704495124244846736, 1.72366327285081231684991845575, 2.48588288085043609780181726987, 4.10592857573591613857576980113, 5.12766076553305769744686596305, 6.15251282623890870959896752930, 6.85191604675138839506117281231, 7.72193331330416100161082137691, 8.445938805563638736868847314563, 9.617333402716541518737312872279, 10.34540162216779262025594233097