# Properties

 Label 2-33-3.2-c2-0-2 Degree $2$ Conductor $33$ Sign $-i$ Analytic cond. $0.899184$ Root an. cond. $0.948253$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 3.31i·2-s + 3·3-s − 7·4-s − 6.63i·5-s + 9.94i·6-s − 8·7-s − 9.94i·8-s + 9·9-s + 22·10-s + 3.31i·11-s − 21·12-s + 4·13-s − 26.5i·14-s − 19.8i·15-s + 5.00·16-s + 13.2i·17-s + ⋯
 L(s)  = 1 + 1.65i·2-s + 3-s − 1.75·4-s − 1.32i·5-s + 1.65i·6-s − 1.14·7-s − 1.24i·8-s + 9-s + 2.20·10-s + 0.301i·11-s − 1.75·12-s + 0.307·13-s − 1.89i·14-s − 1.32i·15-s + 0.312·16-s + 0.780i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$33$$    =    $$3 \cdot 11$$ Sign: $-i$ Analytic conductor: $$0.899184$$ Root analytic conductor: $$0.948253$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{33} (23, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 33,\ (\ :1),\ -i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.813379 + 0.813379i$$ $$L(\frac12)$$ $$\approx$$ $$0.813379 + 0.813379i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 3T$$
11 $$1 - 3.31iT$$
good2 $$1 - 3.31iT - 4T^{2}$$
5 $$1 + 6.63iT - 25T^{2}$$
7 $$1 + 8T + 49T^{2}$$
13 $$1 - 4T + 169T^{2}$$
17 $$1 - 13.2iT - 289T^{2}$$
19 $$1 + 6T + 361T^{2}$$
23 $$1 - 6.63iT - 529T^{2}$$
29 $$1 + 39.7iT - 841T^{2}$$
31 $$1 + 26T + 961T^{2}$$
37 $$1 - 30T + 1.36e3T^{2}$$
41 $$1 - 13.2iT - 1.68e3T^{2}$$
43 $$1 - 42T + 1.84e3T^{2}$$
47 $$1 - 86.2iT - 2.20e3T^{2}$$
53 $$1 + 59.6iT - 2.80e3T^{2}$$
59 $$1 - 66.3iT - 3.48e3T^{2}$$
61 $$1 - 12T + 3.72e3T^{2}$$
67 $$1 - 2T + 4.48e3T^{2}$$
71 $$1 + 59.6iT - 5.04e3T^{2}$$
73 $$1 + 74T + 5.32e3T^{2}$$
79 $$1 + 40T + 6.24e3T^{2}$$
83 $$1 + 39.7iT - 6.88e3T^{2}$$
89 $$1 + 119. iT - 7.92e3T^{2}$$
97 $$1 - 62T + 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

−16.39340489348794926799824590827, −15.78213262934442172356563398244, −14.71618178752052947563904800675, −13.34101850161211493925304591171, −12.77656968448705242970921055972, −9.625344277890424690454082788157, −8.782121035391266122095343451094, −7.68385747545354478235407501398, −6.11198907827616590669215168131, −4.29685542700832488628962029342, 2.68129674928404865106074845948, 3.62327587915032566364728411320, 6.93970335167476522991480177841, 9.026354364107742844429786346354, 10.07999574161445947999091575522, 11.00398082884895023266604412421, 12.57513390229127967014706317425, 13.56469871033383997019074328678, 14.56671741647811408569175428269, 16.01398418524725385160950626319