# Properties

 Label 2-3332-3332.3263-c0-0-2 Degree $2$ Conductor $3332$ Sign $0.926 + 0.375i$ Analytic cond. $1.66288$ Root an. cond. $1.28952$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.900 − 0.433i)2-s + (0.0990 + 0.433i)3-s + (0.623 + 0.781i)4-s + (0.0990 − 0.433i)6-s + (−0.222 − 0.974i)7-s + (−0.222 − 0.974i)8-s + (0.722 − 0.347i)9-s + (1.62 + 0.781i)11-s + (−0.277 + 0.347i)12-s + (1.62 + 0.781i)13-s + (−0.222 + 0.974i)14-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 0.801·18-s + (0.400 − 0.193i)21-s + (−1.12 − 1.40i)22-s + ⋯
 L(s)  = 1 + (−0.900 − 0.433i)2-s + (0.0990 + 0.433i)3-s + (0.623 + 0.781i)4-s + (0.0990 − 0.433i)6-s + (−0.222 − 0.974i)7-s + (−0.222 − 0.974i)8-s + (0.722 − 0.347i)9-s + (1.62 + 0.781i)11-s + (−0.277 + 0.347i)12-s + (1.62 + 0.781i)13-s + (−0.222 + 0.974i)14-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 0.801·18-s + (0.400 − 0.193i)21-s + (−1.12 − 1.40i)22-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3332$$    =    $$2^{2} \cdot 7^{2} \cdot 17$$ Sign: $0.926 + 0.375i$ Analytic conductor: $$1.66288$$ Root analytic conductor: $$1.28952$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3332} (3263, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3332,\ (\ :0),\ 0.926 + 0.375i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.074677317$$ $$L(\frac12)$$ $$\approx$$ $$1.074677317$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.900 + 0.433i)T$$
7 $$1 + (0.222 + 0.974i)T$$
17 $$1 + (-0.623 + 0.781i)T$$
good3 $$1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2}$$
5 $$1 + (0.900 - 0.433i)T^{2}$$
11 $$1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2}$$
13 $$1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2}$$
19 $$1 - T^{2}$$
23 $$1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2}$$
29 $$1 + (0.222 + 0.974i)T^{2}$$
31 $$1 + 0.445T + T^{2}$$
37 $$1 + (0.222 + 0.974i)T^{2}$$
41 $$1 + (0.900 - 0.433i)T^{2}$$
43 $$1 + (0.900 + 0.433i)T^{2}$$
47 $$1 + (-0.623 - 0.781i)T^{2}$$
53 $$1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2}$$
59 $$1 + (0.900 + 0.433i)T^{2}$$
61 $$1 + (0.222 + 0.974i)T^{2}$$
67 $$1 - T^{2}$$
71 $$1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2}$$
73 $$1 + (-0.623 + 0.781i)T^{2}$$
79 $$1 + 1.80T + T^{2}$$
83 $$1 + (-0.623 + 0.781i)T^{2}$$
89 $$1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2}$$
97 $$1 - T^{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

−9.064367908270269060946305347559, −8.152707662710964097548883423395, −7.26179947740138308019505865352, −6.70952275722340195837608770161, −6.16430894839717575852009848330, −4.38062174021191912654306776561, −3.99028868104225092765684075007, −3.41001227877780259508319474166, −1.82250609616824249987916364299, −1.10940514060402664446714313459, 1.27612303701564556537178517596, 1.81553218313765028260279962571, 3.25706194755196163332829296173, 4.04689507810119786006619538020, 5.71536426371494847239119233101, 5.88694964728132425346472724850, 6.56127637021715190624354710696, 7.54044513459968227206675373740, 8.259520707283768019916818097614, 8.629506462096311933827393236719