# Properties

 Label 2-3267-11.6-c0-0-3 Degree $2$ Conductor $3267$ Sign $0.292 + 0.956i$ Analytic cond. $1.63044$ Root an. cond. $1.27688$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.809 + 0.587i)4-s + (−0.304 − 0.418i)7-s + (−1.83 + 0.596i)13-s + (0.309 − 0.951i)16-s + (0.831 − 1.14i)19-s + (0.809 + 0.587i)25-s + (0.492 + 0.159i)28-s + (−0.535 − 1.64i)31-s − 1.41i·43-s + (0.226 − 0.696i)49-s + (1.13 − 1.56i)52-s + (−1.34 − 0.437i)61-s + (0.309 + 0.951i)64-s + 1.73·67-s + (−1.13 − 1.56i)73-s + ⋯
 L(s)  = 1 + (−0.809 + 0.587i)4-s + (−0.304 − 0.418i)7-s + (−1.83 + 0.596i)13-s + (0.309 − 0.951i)16-s + (0.831 − 1.14i)19-s + (0.809 + 0.587i)25-s + (0.492 + 0.159i)28-s + (−0.535 − 1.64i)31-s − 1.41i·43-s + (0.226 − 0.696i)49-s + (1.13 − 1.56i)52-s + (−1.34 − 0.437i)61-s + (0.309 + 0.951i)64-s + 1.73·67-s + (−1.13 − 1.56i)73-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3267$$    =    $$3^{3} \cdot 11^{2}$$ Sign: $0.292 + 0.956i$ Analytic conductor: $$1.63044$$ Root analytic conductor: $$1.27688$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3267} (2998, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3267,\ (\ :0),\ 0.292 + 0.956i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6441723538$$ $$L(\frac12)$$ $$\approx$$ $$0.6441723538$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
11 $$1$$
good2 $$1 + (0.809 - 0.587i)T^{2}$$
5 $$1 + (-0.809 - 0.587i)T^{2}$$
7 $$1 + (0.304 + 0.418i)T + (-0.309 + 0.951i)T^{2}$$
13 $$1 + (1.83 - 0.596i)T + (0.809 - 0.587i)T^{2}$$
17 $$1 + (0.809 + 0.587i)T^{2}$$
19 $$1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2}$$
23 $$1 + T^{2}$$
29 $$1 + (-0.309 + 0.951i)T^{2}$$
31 $$1 + (0.535 + 1.64i)T + (-0.809 + 0.587i)T^{2}$$
37 $$1 + (0.309 - 0.951i)T^{2}$$
41 $$1 + (-0.309 - 0.951i)T^{2}$$
43 $$1 + 1.41iT - T^{2}$$
47 $$1 + (0.309 + 0.951i)T^{2}$$
53 $$1 + (-0.809 + 0.587i)T^{2}$$
59 $$1 + (0.309 - 0.951i)T^{2}$$
61 $$1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2}$$
67 $$1 - 1.73T + T^{2}$$
71 $$1 + (-0.809 - 0.587i)T^{2}$$
73 $$1 + (1.13 + 1.56i)T + (-0.309 + 0.951i)T^{2}$$
79 $$1 + (-1.83 + 0.596i)T + (0.809 - 0.587i)T^{2}$$
83 $$1 + (0.809 + 0.587i)T^{2}$$
89 $$1 + T^{2}$$
97 $$1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$