# Properties

 Label 2-483-161.160-c1-0-9 Degree $2$ Conductor $483$ Sign $0.306 - 0.951i$ 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 − 2.11·2-s + i·3-s + 2.47·4-s − 1.98·5-s − 2.11i·6-s + (2.62 + 0.302i)7-s − 1.00·8-s − 9-s + 4.20·10-s + 2.93i·11-s + 2.47i·12-s − 4.75i·13-s + (−5.55 − 0.638i)14-s − 1.98i·15-s − 2.83·16-s + 0.444·17-s + ⋯
 L(s)  = 1 − 1.49·2-s + 0.577i·3-s + 1.23·4-s − 0.889·5-s − 0.863i·6-s + (0.993 + 0.114i)7-s − 0.353·8-s − 0.333·9-s + 1.33·10-s + 0.883i·11-s + 0.713i·12-s − 1.31i·13-s + (−1.48 − 0.170i)14-s − 0.513i·15-s − 0.707·16-s + 0.107·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$483$$    =    $$3 \cdot 7 \cdot 23$$ Sign: $0.306 - 0.951i$ Analytic conductor: $$3.85677$$ Root analytic conductor: $$1.96386$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{483} (160, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 483,\ (\ :1/2),\ 0.306 - 0.951i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.481495 + 0.350675i$$ $$L(\frac12)$$ $$\approx$$ $$0.481495 + 0.350675i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - iT$$
7 $$1 + (-2.62 - 0.302i)T$$
23 $$1 + (-4.70 - 0.940i)T$$
good2 $$1 + 2.11T + 2T^{2}$$
5 $$1 + 1.98T + 5T^{2}$$
11 $$1 - 2.93iT - 11T^{2}$$
13 $$1 + 4.75iT - 13T^{2}$$
17 $$1 - 0.444T + 17T^{2}$$
19 $$1 - 6.30T + 19T^{2}$$
29 $$1 + 2.58T + 29T^{2}$$
31 $$1 - 6.35iT - 31T^{2}$$
37 $$1 - 6.69iT - 37T^{2}$$
41 $$1 + 10.5iT - 41T^{2}$$
43 $$1 - 12.6iT - 43T^{2}$$
47 $$1 - 5.66iT - 47T^{2}$$
53 $$1 - 10.6iT - 53T^{2}$$
59 $$1 - 6.39iT - 59T^{2}$$
61 $$1 - 3.42T + 61T^{2}$$
67 $$1 - 9.73iT - 67T^{2}$$
71 $$1 - 8.75T + 71T^{2}$$
73 $$1 - 0.926iT - 73T^{2}$$
79 $$1 + 8.86iT - 79T^{2}$$
83 $$1 - 4.81T + 83T^{2}$$
89 $$1 + 0.0511T + 89T^{2}$$
97 $$1 + 17.4T + 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

−10.94912724754416127059413513161, −10.17937731675560034052309081433, −9.373062197100061983220838060654, −8.469128937354522422618243371322, −7.72099758892461295783949485713, −7.24361849621347274936598983000, −5.41441365764304878457384592242, −4.48160811016251598773677905105, −2.99338532709885610225481468324, −1.20636896531278492666254011965, 0.72835674730361729254504643229, 2.02612145654218513515895664373, 3.79478755138396037494679445279, 5.20095936643581447675814062816, 6.72457421878497614519138859921, 7.52008377908606454165767667870, 8.085248215520002949618656714084, 8.866290242562923305180672098505, 9.666560545557179650723549892265, 11.06605210129362824016005921116