# Properties

 Label 2-33e2-3.2-c2-0-52 Degree $2$ Conductor $1089$ Sign $0.577 + 0.816i$ Analytic cond. $29.6731$ Root an. cond. $5.44730$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 2.99i·2-s − 4.96·4-s − 1.61i·5-s − 3.32·7-s − 2.90i·8-s + 4.82·10-s + 7.56·13-s − 9.97i·14-s − 11.1·16-s + 28.3i·17-s − 26.0·19-s + 8.00i·20-s − 19.5i·23-s + 22.4·25-s + 22.6i·26-s + ⋯
 L(s)  = 1 + 1.49i·2-s − 1.24·4-s − 0.322i·5-s − 0.475·7-s − 0.362i·8-s + 0.482·10-s + 0.581·13-s − 0.712i·14-s − 0.698·16-s + 1.66i·17-s − 1.37·19-s + 0.400i·20-s − 0.850i·23-s + 0.896·25-s + 0.871i·26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1089$$    =    $$3^{2} \cdot 11^{2}$$ Sign: $0.577 + 0.816i$ Analytic conductor: $$29.6731$$ Root analytic conductor: $$5.44730$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{1089} (485, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1089,\ (\ :1),\ 0.577 + 0.816i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.1300813558$$ $$L(\frac12)$$ $$\approx$$ $$0.1300813558$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
11 $$1$$
good2 $$1 - 2.99iT - 4T^{2}$$
5 $$1 + 1.61iT - 25T^{2}$$
7 $$1 + 3.32T + 49T^{2}$$
13 $$1 - 7.56T + 169T^{2}$$
17 $$1 - 28.3iT - 289T^{2}$$
19 $$1 + 26.0T + 361T^{2}$$
23 $$1 + 19.5iT - 529T^{2}$$
29 $$1 + 2.15iT - 841T^{2}$$
31 $$1 + 9.22T + 961T^{2}$$
37 $$1 + 67.5T + 1.36e3T^{2}$$
41 $$1 + 29.0iT - 1.68e3T^{2}$$
43 $$1 - 0.719T + 1.84e3T^{2}$$
47 $$1 + 24.5iT - 2.20e3T^{2}$$
53 $$1 + 5.79iT - 2.80e3T^{2}$$
59 $$1 + 73.9iT - 3.48e3T^{2}$$
61 $$1 - 72.1T + 3.72e3T^{2}$$
67 $$1 + 79.8T + 4.48e3T^{2}$$
71 $$1 + 107. iT - 5.04e3T^{2}$$
73 $$1 + 90.3T + 5.32e3T^{2}$$
79 $$1 - 114.T + 6.24e3T^{2}$$
83 $$1 + 125. iT - 6.88e3T^{2}$$
89 $$1 - 83.3iT - 7.92e3T^{2}$$
97 $$1 + 78.0T + 9.40e3T^{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

−9.011625174692596310523846086741, −8.598373663320143515782313282050, −7.953317202931829876805557146665, −6.72911448298558300620854565757, −6.41501443437068909162527650157, −5.51295871063408895844078455077, −4.55296651669637571134679941327, −3.60675870354707340129832086711, −1.93392039720923314851924845695, −0.03996517539602733269553786772, 1.31803972594885055896761285253, 2.55813658672398936705399678169, 3.26913201978142108102621285522, 4.20828631474685970801105989261, 5.25021326262632759999494527048, 6.53257753829777911327376265159, 7.20441566629093808690673997840, 8.569230634568019519458201477374, 9.233089155357903755053536606624, 9.991971839407682583799620825553