# Properties

 Degree $2$ Conductor $33$ Sign $-0.830 - 0.557i$ Motivic weight $4$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 5.58i·2-s − 5.19·3-s − 15.1·4-s − 29.7·5-s + 29.0i·6-s + 12.9i·7-s − 4.47i·8-s + 27·9-s + 166. i·10-s + (−100. − 67.4i)11-s + 78.9·12-s + 36.6i·13-s + 72.2·14-s + 154.·15-s − 268.·16-s − 464. i·17-s + ⋯
 L(s)  = 1 − 1.39i·2-s − 0.577·3-s − 0.949·4-s − 1.18·5-s + 0.806i·6-s + 0.264i·7-s − 0.0698i·8-s + 0.333·9-s + 1.66i·10-s + (−0.830 − 0.557i)11-s + 0.548·12-s + 0.216i·13-s + 0.368·14-s + 0.687·15-s − 1.04·16-s − 1.60i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 - 0.557i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$33$$    =    $$3 \cdot 11$$ Sign: $-0.830 - 0.557i$ Motivic weight: $$4$$ Character: $\chi_{33} (10, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 33,\ (\ :2),\ -0.830 - 0.557i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$0.157914 + 0.518171i$$ $$L(\frac12)$$ $$\approx$$ $$0.157914 + 0.518171i$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + 5.19T$$
11 $$1 + (100. + 67.4i)T$$
good2 $$1 + 5.58iT - 16T^{2}$$
5 $$1 + 29.7T + 625T^{2}$$
7 $$1 - 12.9iT - 2.40e3T^{2}$$
13 $$1 - 36.6iT - 2.85e4T^{2}$$
17 $$1 + 464. iT - 8.35e4T^{2}$$
19 $$1 + 327. iT - 1.30e5T^{2}$$
23 $$1 - 396.T + 2.79e5T^{2}$$
29 $$1 - 1.15e3iT - 7.07e5T^{2}$$
31 $$1 - 437.T + 9.23e5T^{2}$$
37 $$1 - 276.T + 1.87e6T^{2}$$
41 $$1 + 2.78e3iT - 2.82e6T^{2}$$
43 $$1 - 1.66e3iT - 3.41e6T^{2}$$
47 $$1 + 1.85e3T + 4.87e6T^{2}$$
53 $$1 - 3.74e3T + 7.89e6T^{2}$$
59 $$1 + 6.57e3T + 1.21e7T^{2}$$
61 $$1 + 5.14e3iT - 1.38e7T^{2}$$
67 $$1 + 3.46e3T + 2.01e7T^{2}$$
71 $$1 + 6.03e3T + 2.54e7T^{2}$$
73 $$1 + 3.58e3iT - 2.83e7T^{2}$$
79 $$1 + 9.71e3iT - 3.89e7T^{2}$$
83 $$1 - 1.18e4iT - 4.74e7T^{2}$$
89 $$1 - 2.58e3T + 6.27e7T^{2}$$
97 $$1 - 1.55e3T + 8.85e7T^{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

−15.50715904263705374030557485149, −13.52910293503173915948083375567, −12.29965043916977721682914821166, −11.47087644907903782264864217241, −10.71211818117809503166568304418, −9.077544503380697826856464630967, −7.22522899829977447971499077824, −4.81129686017761205338950136874, −3.04645909623046377122854596137, −0.41487617480888194126538289288, 4.37054655346817050757255810574, 5.96618750392371359288711092566, 7.43619867260999712851446398765, 8.238345459531409391782889250902, 10.42007192871083289145996850636, 11.81217244566027900373039877889, 13.16735446576011077642377366865, 14.94399648930167652512609377375, 15.45040218148692674548079563583, 16.54740223139641928593093745542