# Properties

 Label 2-3332-476.131-c0-0-1 Degree $2$ Conductor $3332$ Sign $0.695 - 0.718i$ 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 + (0.991 + 0.130i)2-s + (0.965 + 0.258i)4-s + (0.835 + 0.732i)5-s + (0.923 + 0.382i)8-s + (−0.608 − 0.793i)9-s + (0.732 + 0.835i)10-s + (−1 + i)13-s + (0.866 + 0.5i)16-s + (0.793 + 0.608i)17-s + (−0.499 − 0.866i)18-s + (0.617 + 0.923i)20-s + (0.0306 + 0.232i)25-s + (−1.12 + 0.860i)26-s + (−0.216 + 1.08i)29-s + (0.793 + 0.608i)32-s + ⋯
 L(s)  = 1 + (0.991 + 0.130i)2-s + (0.965 + 0.258i)4-s + (0.835 + 0.732i)5-s + (0.923 + 0.382i)8-s + (−0.608 − 0.793i)9-s + (0.732 + 0.835i)10-s + (−1 + i)13-s + (0.866 + 0.5i)16-s + (0.793 + 0.608i)17-s + (−0.499 − 0.866i)18-s + (0.617 + 0.923i)20-s + (0.0306 + 0.232i)25-s + (−1.12 + 0.860i)26-s + (−0.216 + 1.08i)29-s + (0.793 + 0.608i)32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3332$$    =    $$2^{2} \cdot 7^{2} \cdot 17$$ Sign: $0.695 - 0.718i$ Analytic conductor: $$1.66288$$ Root analytic conductor: $$1.28952$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3332} (607, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3332,\ (\ :0),\ 0.695 - 0.718i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.655765195$$ $$L(\frac12)$$ $$\approx$$ $$2.655765195$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.991 - 0.130i)T$$
7 $$1$$
17 $$1 + (-0.793 - 0.608i)T$$
good3 $$1 + (0.608 + 0.793i)T^{2}$$
5 $$1 + (-0.835 - 0.732i)T + (0.130 + 0.991i)T^{2}$$
11 $$1 + (-0.991 - 0.130i)T^{2}$$
13 $$1 + (1 - i)T - iT^{2}$$
19 $$1 + (0.965 + 0.258i)T^{2}$$
23 $$1 + (-0.608 + 0.793i)T^{2}$$
29 $$1 + (0.216 - 1.08i)T + (-0.923 - 0.382i)T^{2}$$
31 $$1 + (-0.608 - 0.793i)T^{2}$$
37 $$1 + (0.108 + 1.65i)T + (-0.991 + 0.130i)T^{2}$$
41 $$1 + (-0.0761 - 0.382i)T + (-0.923 + 0.382i)T^{2}$$
43 $$1 + (0.707 + 0.707i)T^{2}$$
47 $$1 + (0.866 - 0.5i)T^{2}$$
53 $$1 + (-1.12 + 1.46i)T + (-0.258 - 0.965i)T^{2}$$
59 $$1 + (-0.965 + 0.258i)T^{2}$$
61 $$1 + (-0.630 + 1.85i)T + (-0.793 - 0.608i)T^{2}$$
67 $$1 + (-0.5 + 0.866i)T^{2}$$
71 $$1 + (-0.382 + 0.923i)T^{2}$$
73 $$1 + (1.57 - 0.534i)T + (0.793 - 0.608i)T^{2}$$
79 $$1 + (0.608 - 0.793i)T^{2}$$
83 $$1 + (0.707 - 0.707i)T^{2}$$
89 $$1 + (0.866 - 0.5i)T^{2}$$
97 $$1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$