# Properties

 Label 2-3332-476.219-c0-0-3 Degree $2$ Conductor $3332$ Sign $0.123 + 0.992i$ 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 + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.0999 − 0.758i)5-s + (0.707 − 0.707i)8-s + (−0.258 − 0.965i)9-s + (−0.0999 − 0.758i)10-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)17-s + (−0.499 − 0.866i)18-s + (−0.292 − 0.707i)20-s + (0.400 + 0.107i)25-s + (0.707 − 0.292i)29-s + (0.258 − 0.965i)32-s + i·34-s + (−0.707 − 0.707i)36-s + (−0.758 − 0.0999i)37-s + ⋯
 L(s)  = 1 + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.0999 − 0.758i)5-s + (0.707 − 0.707i)8-s + (−0.258 − 0.965i)9-s + (−0.0999 − 0.758i)10-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)17-s + (−0.499 − 0.866i)18-s + (−0.292 − 0.707i)20-s + (0.400 + 0.107i)25-s + (0.707 − 0.292i)29-s + (0.258 − 0.965i)32-s + i·34-s + (−0.707 − 0.707i)36-s + (−0.758 − 0.0999i)37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3332$$    =    $$2^{2} \cdot 7^{2} \cdot 17$$ Sign: $0.123 + 0.992i$ Analytic conductor: $$1.66288$$ Root analytic conductor: $$1.28952$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3332} (1647, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3332,\ (\ :0),\ 0.123 + 0.992i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.340239883$$ $$L(\frac12)$$ $$\approx$$ $$2.340239883$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.965 + 0.258i)T$$
7 $$1$$
17 $$1 + (0.258 - 0.965i)T$$
good3 $$1 + (0.258 + 0.965i)T^{2}$$
5 $$1 + (-0.0999 + 0.758i)T + (-0.965 - 0.258i)T^{2}$$
11 $$1 + (-0.965 + 0.258i)T^{2}$$
13 $$1 - T^{2}$$
19 $$1 + (-0.866 + 0.5i)T^{2}$$
23 $$1 + (0.258 - 0.965i)T^{2}$$
29 $$1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2}$$
31 $$1 + (0.258 + 0.965i)T^{2}$$
37 $$1 + (0.758 + 0.0999i)T + (0.965 + 0.258i)T^{2}$$
41 $$1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2}$$
43 $$1 + iT^{2}$$
47 $$1 + (-0.5 - 0.866i)T^{2}$$
53 $$1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2}$$
59 $$1 + (-0.866 - 0.5i)T^{2}$$
61 $$1 + (-1.46 + 1.12i)T + (0.258 - 0.965i)T^{2}$$
67 $$1 + (0.5 - 0.866i)T^{2}$$
71 $$1 + (0.707 - 0.707i)T^{2}$$
73 $$1 + (0.607 + 0.465i)T + (0.258 + 0.965i)T^{2}$$
79 $$1 + (0.258 - 0.965i)T^{2}$$
83 $$1 - iT^{2}$$
89 $$1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2}$$
97 $$1 + (0.707 - 0.292i)T + (0.707 - 0.707i)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

−8.718121995825093886757082408793, −7.88705168490309106426204937911, −6.78410677624725477064403623972, −6.34171621573562970195706942602, −5.46044152287244802555162706659, −4.81926753746392091228970635769, −3.91812786811348459960255313787, −3.28046304674850454497978475998, −2.11276847089133480192130302580, −1.05740136694526897986230506952, 1.86137447689934077097655311697, 2.76597752983309755168459772768, 3.34899124692051940482125257461, 4.56424734493325392479743343890, 5.09451813857499216911187577111, 5.89533588073568944849463367848, 6.92324425209449342205079439440, 7.05828617196057181011390131756, 8.147375211143371882560108155336, 8.664523605165466202877320599531