# Properties

 Label 2-33e2-33.32-c1-0-19 Degree $2$ Conductor $1089$ Sign $0.384 + 0.923i$ Analytic cond. $8.69570$ Root an. cond. $2.94884$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Learn more

## Dirichlet series

 L(s)  = 1 − 0.688·2-s − 1.52·4-s − 0.0401i·5-s + 0.246i·7-s + 2.42·8-s + 0.0276i·10-s − 2.30i·13-s − 0.169i·14-s + 1.38·16-s − 4.27·17-s + 6.20i·19-s + 0.0612i·20-s − 6.79i·23-s + 4.99·25-s + 1.58i·26-s + ⋯
 L(s)  = 1 − 0.486·2-s − 0.763·4-s − 0.0179i·5-s + 0.0932i·7-s + 0.858·8-s + 0.00872i·10-s − 0.638i·13-s − 0.0454i·14-s + 0.345·16-s − 1.03·17-s + 1.42i·19-s + 0.0136i·20-s − 1.41i·23-s + 0.999·25-s + 0.310i·26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1089$$    =    $$3^{2} \cdot 11^{2}$$ Sign: $0.384 + 0.923i$ Analytic conductor: $$8.69570$$ Root analytic conductor: $$2.94884$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1089} (1088, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1089,\ (\ :1/2),\ 0.384 + 0.923i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.7941804466$$ $$L(\frac12)$$ $$\approx$$ $$0.7941804466$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
11 $$1$$
good2 $$1 + 0.688T + 2T^{2}$$
5 $$1 + 0.0401iT - 5T^{2}$$
7 $$1 - 0.246iT - 7T^{2}$$
13 $$1 + 2.30iT - 13T^{2}$$
17 $$1 + 4.27T + 17T^{2}$$
19 $$1 - 6.20iT - 19T^{2}$$
23 $$1 + 6.79iT - 23T^{2}$$
29 $$1 + 5.59T + 29T^{2}$$
31 $$1 - 4.79T + 31T^{2}$$
37 $$1 + 4.03T + 37T^{2}$$
41 $$1 - 9.60T + 41T^{2}$$
43 $$1 + 1.03iT - 43T^{2}$$
47 $$1 + 11.1iT - 47T^{2}$$
53 $$1 + 8.96iT - 53T^{2}$$
59 $$1 + 2.78iT - 59T^{2}$$
61 $$1 + 8.48iT - 61T^{2}$$
67 $$1 - 7.94T + 67T^{2}$$
71 $$1 + 3.32iT - 71T^{2}$$
73 $$1 + 11.8iT - 73T^{2}$$
79 $$1 + 3.01iT - 79T^{2}$$
83 $$1 + 5.29T + 83T^{2}$$
89 $$1 + 8.54iT - 89T^{2}$$
97 $$1 + 3.02T + 97T^{2}$$
show more
show less
$$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

−9.709259270341238189946421873510, −8.740319946165314148553589741992, −8.341813322465757819307096544023, −7.40735584071940610329828522591, −6.37408112398852213898326472885, −5.35012247395157256996385435979, −4.48285572926991813837212151585, −3.53231076987696099132894774539, −2.07359307152150981892691724188, −0.51408885130907543948359990438, 1.11819915132474521782161659814, 2.61713481083898514510419594548, 4.03858853264800273785022775858, 4.69493841996800004765734577326, 5.71183822392350844525876943205, 6.93045106611899315665111024350, 7.53239586995472275178768725296, 8.661944624461309309707127280918, 9.151847031324253111812193096987, 9.749590656697379537345782249528