# Properties

 Label 2-3267-297.14-c0-0-0 Degree $2$ Conductor $3267$ Sign $-0.988 - 0.150i$ 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 + (−0.990 + 0.139i)3-s + (0.0348 + 0.999i)4-s + (−0.663 + 0.165i)5-s + (0.961 − 0.275i)9-s + (−0.173 − 0.984i)12-s + (0.634 − 0.256i)15-s + (−0.997 + 0.0697i)16-s + (−0.188 − 0.657i)20-s + (0.592 + 1.62i)23-s + (−0.469 + 0.249i)25-s + (−0.913 + 0.406i)27-s + (−0.194 + 0.287i)31-s + (0.309 + 0.951i)36-s + (1.25 + 1.39i)37-s + (−0.592 + 0.342i)45-s + ⋯
 L(s)  = 1 + (−0.990 + 0.139i)3-s + (0.0348 + 0.999i)4-s + (−0.663 + 0.165i)5-s + (0.961 − 0.275i)9-s + (−0.173 − 0.984i)12-s + (0.634 − 0.256i)15-s + (−0.997 + 0.0697i)16-s + (−0.188 − 0.657i)20-s + (0.592 + 1.62i)23-s + (−0.469 + 0.249i)25-s + (−0.913 + 0.406i)27-s + (−0.194 + 0.287i)31-s + (0.309 + 0.951i)36-s + (1.25 + 1.39i)37-s + (−0.592 + 0.342i)45-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3267$$    =    $$3^{3} \cdot 11^{2}$$ Sign: $-0.988 - 0.150i$ Analytic conductor: $$1.63044$$ Root analytic conductor: $$1.27688$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3267} (608, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3267,\ (\ :0),\ -0.988 - 0.150i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.4309923862$$ $$L(\frac12)$$ $$\approx$$ $$0.4309923862$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (0.990 - 0.139i)T$$
11 $$1$$
good2 $$1 + (-0.0348 - 0.999i)T^{2}$$
5 $$1 + (0.663 - 0.165i)T + (0.882 - 0.469i)T^{2}$$
7 $$1 + (-0.241 - 0.970i)T^{2}$$
13 $$1 + (-0.615 + 0.788i)T^{2}$$
17 $$1 + (-0.669 + 0.743i)T^{2}$$
19 $$1 + (-0.104 - 0.994i)T^{2}$$
23 $$1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2}$$
29 $$1 + (-0.961 - 0.275i)T^{2}$$
31 $$1 + (0.194 - 0.287i)T + (-0.374 - 0.927i)T^{2}$$
37 $$1 + (-1.25 - 1.39i)T + (-0.104 + 0.994i)T^{2}$$
41 $$1 + (-0.961 + 0.275i)T^{2}$$
43 $$1 + (0.173 - 0.984i)T^{2}$$
47 $$1 + (1.28 + 0.0448i)T + (0.997 + 0.0697i)T^{2}$$
53 $$1 + (1.15 + 1.59i)T + (-0.309 + 0.951i)T^{2}$$
59 $$1 + (1.04 + 1.67i)T + (-0.438 + 0.898i)T^{2}$$
61 $$1 + (-0.374 + 0.927i)T^{2}$$
67 $$1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2}$$
71 $$1 + (0.522 + 1.17i)T + (-0.669 + 0.743i)T^{2}$$
73 $$1 + (0.913 - 0.406i)T^{2}$$
79 $$1 + (0.0348 + 0.999i)T^{2}$$
83 $$1 + (0.615 + 0.788i)T^{2}$$
89 $$1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}$$
97 $$1 + (0.370 - 1.48i)T + (-0.882 - 0.469i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$