# Properties

 Label 2-1083-1.1-c5-0-28 Degree $2$ Conductor $1083$ Sign $1$ Analytic cond. $173.695$ Root an. cond. $13.1793$ Motivic weight $5$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 6·2-s − 9·3-s + 4·4-s + 6·5-s − 54·6-s − 40·7-s − 168·8-s + 81·9-s + 36·10-s − 564·11-s − 36·12-s − 638·13-s − 240·14-s − 54·15-s − 1.13e3·16-s + 882·17-s + 486·18-s + 24·20-s + 360·21-s − 3.38e3·22-s − 840·23-s + 1.51e3·24-s − 3.08e3·25-s − 3.82e3·26-s − 729·27-s − 160·28-s − 4.63e3·29-s + ⋯
 L(s)  = 1 + 1.06·2-s − 0.577·3-s + 1/8·4-s + 0.107·5-s − 0.612·6-s − 0.308·7-s − 0.928·8-s + 1/3·9-s + 0.113·10-s − 1.40·11-s − 0.0721·12-s − 1.04·13-s − 0.327·14-s − 0.0619·15-s − 1.10·16-s + 0.740·17-s + 0.353·18-s + 0.0134·20-s + 0.178·21-s − 1.49·22-s − 0.331·23-s + 0.535·24-s − 0.988·25-s − 1.11·26-s − 0.192·27-s − 0.0385·28-s − 1.02·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1083$$    =    $$3 \cdot 19^{2}$$ Sign: $1$ Analytic conductor: $$173.695$$ Root analytic conductor: $$13.1793$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1083,\ (\ :5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.8028987812$$ $$L(\frac12)$$ $$\approx$$ $$0.8028987812$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + p^{2} T$$
19 $$1$$
good2 $$1 - 3 p T + p^{5} T^{2}$$
5 $$1 - 6 T + p^{5} T^{2}$$
7 $$1 + 40 T + p^{5} T^{2}$$
11 $$1 + 564 T + p^{5} T^{2}$$
13 $$1 + 638 T + p^{5} T^{2}$$
17 $$1 - 882 T + p^{5} T^{2}$$
23 $$1 + 840 T + p^{5} T^{2}$$
29 $$1 + 4638 T + p^{5} T^{2}$$
31 $$1 + 4400 T + p^{5} T^{2}$$
37 $$1 - 2410 T + p^{5} T^{2}$$
41 $$1 - 6870 T + p^{5} T^{2}$$
43 $$1 - 9644 T + p^{5} T^{2}$$
47 $$1 + 18672 T + p^{5} T^{2}$$
53 $$1 + 33750 T + p^{5} T^{2}$$
59 $$1 - 18084 T + p^{5} T^{2}$$
61 $$1 - 39758 T + p^{5} T^{2}$$
67 $$1 - 23068 T + p^{5} T^{2}$$
71 $$1 - 4248 T + p^{5} T^{2}$$
73 $$1 + 41110 T + p^{5} T^{2}$$
79 $$1 + 21920 T + p^{5} T^{2}$$
83 $$1 - 82452 T + p^{5} T^{2}$$
89 $$1 - 94086 T + p^{5} T^{2}$$
97 $$1 + 49442 T + p^{5} 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

−9.496836595909973790585042390430, −8.080345409244966479626722466801, −7.38917060485400188180780658399, −6.27266724183700216894881470675, −5.48241881589994775127136969577, −5.05460062120767708136580067217, −4.02609871485109633885335604064, −3.06423705626129388363328242660, −2.07827300670332858444113879625, −0.31498196263593192206757309694, 0.31498196263593192206757309694, 2.07827300670332858444113879625, 3.06423705626129388363328242660, 4.02609871485109633885335604064, 5.05460062120767708136580067217, 5.48241881589994775127136969577, 6.27266724183700216894881470675, 7.38917060485400188180780658399, 8.080345409244966479626722466801, 9.496836595909973790585042390430