# Properties

 Label 2-3332-476.191-c0-0-1 Degree $2$ Conductor $3332$ Sign $0.510 + 0.859i$ 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 + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.366 − 1.36i)5-s − 0.999i·8-s + (0.866 + 0.5i)9-s + (−0.366 + 1.36i)10-s + 2·13-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.499 − 0.866i)18-s + (0.999 − 0.999i)20-s + (−0.866 + 0.5i)25-s + (−1.73 − i)26-s + (−1 + i)29-s + (0.866 − 0.499i)32-s + ⋯
 L(s)  = 1 + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.366 − 1.36i)5-s − 0.999i·8-s + (0.866 + 0.5i)9-s + (−0.366 + 1.36i)10-s + 2·13-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.499 − 0.866i)18-s + (0.999 − 0.999i)20-s + (−0.866 + 0.5i)25-s + (−1.73 − i)26-s + (−1 + i)29-s + (0.866 − 0.499i)32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3332$$    =    $$2^{2} \cdot 7^{2} \cdot 17$$ Sign: $0.510 + 0.859i$ Analytic conductor: $$1.66288$$ Root analytic conductor: $$1.28952$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3332} (667, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3332,\ (\ :0),\ 0.510 + 0.859i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.9685260811$$ $$L(\frac12)$$ $$\approx$$ $$0.9685260811$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.866 + 0.5i)T$$
7 $$1$$
17 $$1 + (-0.866 + 0.5i)T$$
good3 $$1 + (-0.866 - 0.5i)T^{2}$$
5 $$1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2}$$
11 $$1 + (0.866 + 0.5i)T^{2}$$
13 $$1 - 2T + T^{2}$$
19 $$1 + (-0.5 - 0.866i)T^{2}$$
23 $$1 + (-0.866 + 0.5i)T^{2}$$
29 $$1 + (1 - i)T - iT^{2}$$
31 $$1 + (-0.866 - 0.5i)T^{2}$$
37 $$1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2}$$
41 $$1 + (-1 - i)T + iT^{2}$$
43 $$1 + T^{2}$$
47 $$1 + (0.5 + 0.866i)T^{2}$$
53 $$1 + (0.5 - 0.866i)T^{2}$$
59 $$1 + (-0.5 + 0.866i)T^{2}$$
61 $$1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2}$$
67 $$1 + (0.5 - 0.866i)T^{2}$$
71 $$1 + iT^{2}$$
73 $$1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2}$$
79 $$1 + (-0.866 + 0.5i)T^{2}$$
83 $$1 + T^{2}$$
89 $$1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}$$
97 $$1 + (1 - i)T - iT^{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

−8.696523585368969429833165058444, −8.099857452273330144927158153607, −7.58101760149600110564752394401, −6.61130599457158151863301787413, −5.65296988350534869298472054807, −4.64947582485251837650653996045, −3.94570576014562021581925710889, −3.09752956401576953025995233556, −1.53683453102130554480168920674, −1.11758881126011125926257269677, 1.11293147012693427219868294174, 2.25003092462770182534077476561, 3.52462170581088793703702568455, 3.99545074357470701238648966509, 5.57187666913145199100515301077, 6.23298662726150483540466471606, 6.68487333782449276333389837327, 7.64051642119053652538365908114, 7.87628332121347762491039638003, 9.005703529698859567353863122746