# Properties

 Label 2-33-33.32-c5-0-0 Degree $2$ Conductor $33$ Sign $-0.469 - 0.882i$ Analytic cond. $5.29266$ Root an. cond. $2.30057$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 3.58·2-s + (0.133 − 15.5i)3-s − 19.1·4-s − 11.8i·5-s + (−0.480 + 55.9i)6-s + 150. i·7-s + 183.·8-s + (−242. − 4.17i)9-s + 42.6i·10-s + (−352. + 191. i)11-s + (−2.56 + 297. i)12-s + 778. i·13-s − 541. i·14-s + (−185. − 1.59i)15-s − 47.1·16-s − 1.25e3·17-s + ⋯
 L(s)  = 1 − 0.634·2-s + (0.00859 − 0.999i)3-s − 0.597·4-s − 0.212i·5-s + (−0.00545 + 0.634i)6-s + 1.16i·7-s + 1.01·8-s + (−0.999 − 0.0171i)9-s + 0.134i·10-s + (−0.878 + 0.477i)11-s + (−0.00513 + 0.597i)12-s + 1.27i·13-s − 0.738i·14-s + (−0.212 − 0.00182i)15-s − 0.0460·16-s − 1.05·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$33$$    =    $$3 \cdot 11$$ Sign: $-0.469 - 0.882i$ Analytic conductor: $$5.29266$$ Root analytic conductor: $$2.30057$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{33} (32, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 33,\ (\ :5/2),\ -0.469 - 0.882i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.124862 + 0.207903i$$ $$L(\frac12)$$ $$\approx$$ $$0.124862 + 0.207903i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-0.133 + 15.5i)T$$
11 $$1 + (352. - 191. i)T$$
good2 $$1 + 3.58T + 32T^{2}$$
5 $$1 + 11.8iT - 3.12e3T^{2}$$
7 $$1 - 150. iT - 1.68e4T^{2}$$
13 $$1 - 778. iT - 3.71e5T^{2}$$
17 $$1 + 1.25e3T + 1.41e6T^{2}$$
19 $$1 + 1.21e3iT - 2.47e6T^{2}$$
23 $$1 + 880. iT - 6.43e6T^{2}$$
29 $$1 + 3.20e3T + 2.05e7T^{2}$$
31 $$1 + 6.44e3T + 2.86e7T^{2}$$
37 $$1 + 9.31e3T + 6.93e7T^{2}$$
41 $$1 + 5.39e3T + 1.15e8T^{2}$$
43 $$1 - 2.08e4iT - 1.47e8T^{2}$$
47 $$1 + 1.12e4iT - 2.29e8T^{2}$$
53 $$1 + 2.39e4iT - 4.18e8T^{2}$$
59 $$1 - 2.29e4iT - 7.14e8T^{2}$$
61 $$1 + 2.72e4iT - 8.44e8T^{2}$$
67 $$1 - 2.02e4T + 1.35e9T^{2}$$
71 $$1 - 5.64e4iT - 1.80e9T^{2}$$
73 $$1 + 1.20e4iT - 2.07e9T^{2}$$
79 $$1 + 1.10e4iT - 3.07e9T^{2}$$
83 $$1 - 5.15e4T + 3.93e9T^{2}$$
89 $$1 - 8.41e4iT - 5.58e9T^{2}$$
97 $$1 - 9.93e4T + 8.58e9T^{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

−16.36291949900243955152896521984, −14.79811757943134527985049793441, −13.42644442750317682722670044555, −12.57337269571818819580876078093, −11.17218953004746087871150696415, −9.238573725356789214494365014968, −8.489776571803492698139998347429, −6.95321199731445064351198618126, −5.06797202657983422753352010887, −2.05634583321655603632335110553, 0.17793631367946754520878291401, 3.67154108912693610473334433957, 5.23869708196363689932294637787, 7.66655399637113535137897411094, 8.914095763520900304078698336044, 10.42683808438370123724284611310, 10.69717170682866839326652435687, 13.10693538077639394781762899150, 14.12595372014904097782333430333, 15.48573640427334074422795067618