# Properties

 Label 2-483-161.17-c1-0-7 Degree $2$ Conductor $483$ Sign $-0.297 - 0.954i$ Analytic cond. $3.85677$ Root an. cond. $1.96386$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−2.78 + 0.265i)2-s + (0.690 + 0.723i)3-s + (5.70 − 1.10i)4-s + (0.0901 + 0.0464i)5-s + (−2.11 − 1.83i)6-s + (2.63 + 0.236i)7-s + (−10.2 + 3.00i)8-s + (−0.0475 + 0.998i)9-s + (−0.263 − 0.105i)10-s + (−0.291 + 3.04i)11-s + (4.73 + 3.37i)12-s + (−5.20 + 0.749i)13-s + (−7.39 + 0.0431i)14-s + (0.0285 + 0.0972i)15-s + (16.8 − 6.74i)16-s + (0.524 + 1.51i)17-s + ⋯
 L(s)  = 1 + (−1.96 + 0.187i)2-s + (0.398 + 0.417i)3-s + (2.85 − 0.550i)4-s + (0.0402 + 0.0207i)5-s + (−0.862 − 0.747i)6-s + (0.996 + 0.0892i)7-s + (−3.61 + 1.06i)8-s + (−0.0158 + 0.332i)9-s + (−0.0831 − 0.0333i)10-s + (−0.0877 + 0.919i)11-s + (1.36 + 0.973i)12-s + (−1.44 + 0.207i)13-s + (−1.97 + 0.0115i)14-s + (0.00737 + 0.0251i)15-s + (4.21 − 1.68i)16-s + (0.127 + 0.367i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$483$$    =    $$3 \cdot 7 \cdot 23$$ Sign: $-0.297 - 0.954i$ Analytic conductor: $$3.85677$$ Root analytic conductor: $$1.96386$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{483} (178, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 483,\ (\ :1/2),\ -0.297 - 0.954i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.369512 + 0.502382i$$ $$L(\frac12)$$ $$\approx$$ $$0.369512 + 0.502382i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-0.690 - 0.723i)T$$
7 $$1 + (-2.63 - 0.236i)T$$
23 $$1 + (-4.54 + 1.53i)T$$
good2 $$1 + (2.78 - 0.265i)T + (1.96 - 0.378i)T^{2}$$
5 $$1 + (-0.0901 - 0.0464i)T + (2.90 + 4.07i)T^{2}$$
11 $$1 + (0.291 - 3.04i)T + (-10.8 - 2.08i)T^{2}$$
13 $$1 + (5.20 - 0.749i)T + (12.4 - 3.66i)T^{2}$$
17 $$1 + (-0.524 - 1.51i)T + (-13.3 + 10.5i)T^{2}$$
19 $$1 + (2.69 - 7.78i)T + (-14.9 - 11.7i)T^{2}$$
29 $$1 + (-2.32 + 2.67i)T + (-4.12 - 28.7i)T^{2}$$
31 $$1 + (-4.80 - 1.16i)T + (27.5 + 14.2i)T^{2}$$
37 $$1 + (2.77 + 0.132i)T + (36.8 + 3.51i)T^{2}$$
41 $$1 + (-1.21 - 1.88i)T + (-17.0 + 37.2i)T^{2}$$
43 $$1 + (2.33 - 7.94i)T + (-36.1 - 23.2i)T^{2}$$
47 $$1 + (-1.46 + 0.845i)T + (23.5 - 40.7i)T^{2}$$
53 $$1 + (3.97 - 5.05i)T + (-12.4 - 51.5i)T^{2}$$
59 $$1 + (2.85 - 7.12i)T + (-42.7 - 40.7i)T^{2}$$
61 $$1 + (-5.20 - 4.96i)T + (2.90 + 60.9i)T^{2}$$
67 $$1 + (2.83 - 2.02i)T + (21.9 - 63.3i)T^{2}$$
71 $$1 + (-2.38 + 5.22i)T + (-46.4 - 53.6i)T^{2}$$
73 $$1 + (1.33 + 6.95i)T + (-67.7 + 27.1i)T^{2}$$
79 $$1 + (-2.21 - 2.82i)T + (-18.6 + 76.7i)T^{2}$$
83 $$1 + (1.82 + 1.17i)T + (34.4 + 75.4i)T^{2}$$
89 $$1 + (0.298 + 1.23i)T + (-79.1 + 40.7i)T^{2}$$
97 $$1 + (3.47 - 2.23i)T + (40.2 - 88.2i)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

−10.71285346951854409759769128560, −10.12358887314646899616499603282, −9.558049775601576986981320986638, −8.467348956432874397105443931041, −7.940669824998943829592995898044, −7.18599303800133494273172537171, −6.01004981614012018069267138156, −4.61487525649586778043271755407, −2.60256664183278287038842459724, −1.67866980181231331260644884944, 0.66520892771403532288609595128, 2.10710315781660920944255610892, 3.03989808489241581253106739517, 5.23436414180836820045336118085, 6.77286628940458329363162434834, 7.35056133414719609302989180178, 8.193409499497860084275610474620, 8.870852063127431757259474582129, 9.597905490910358668113765575727, 10.65062569288815460045674456175