# Properties

 Degree $2$ Conductor $3264$ Sign $-0.970 + 0.242i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·3-s + 4i·7-s − 9-s + 4i·11-s − 2·13-s + (1 + 4i)17-s − 4·19-s − 4·21-s + 4i·23-s + 5·25-s − i·27-s + 4i·31-s − 4·33-s − 8i·37-s − 2i·39-s + ⋯
 L(s)  = 1 + 0.577i·3-s + 1.51i·7-s − 0.333·9-s + 1.20i·11-s − 0.554·13-s + (0.242 + 0.970i)17-s − 0.917·19-s − 0.872·21-s + 0.834i·23-s + 25-s − 0.192i·27-s + 0.718i·31-s − 0.696·33-s − 1.31i·37-s − 0.320i·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3264$$    =    $$2^{6} \cdot 3 \cdot 17$$ Sign: $-0.970 + 0.242i$ Motivic weight: $$1$$ Character: $\chi_{3264} (577, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3264,\ (\ :1/2),\ -0.970 + 0.242i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.114495186$$ $$L(\frac12)$$ $$\approx$$ $$1.114495186$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - iT$$
17 $$1 + (-1 - 4i)T$$
good5 $$1 - 5T^{2}$$
7 $$1 - 4iT - 7T^{2}$$
11 $$1 - 4iT - 11T^{2}$$
13 $$1 + 2T + 13T^{2}$$
19 $$1 + 4T + 19T^{2}$$
23 $$1 - 4iT - 23T^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 - 4iT - 31T^{2}$$
37 $$1 + 8iT - 37T^{2}$$
41 $$1 + 8iT - 41T^{2}$$
43 $$1 + 4T + 43T^{2}$$
47 $$1 - 8T + 47T^{2}$$
53 $$1 + 6T + 53T^{2}$$
59 $$1 + 12T + 59T^{2}$$
61 $$1 - 8iT - 61T^{2}$$
67 $$1 - 12T + 67T^{2}$$
71 $$1 + 12iT - 71T^{2}$$
73 $$1 - 73T^{2}$$
79 $$1 - 4iT - 79T^{2}$$
83 $$1 - 12T + 83T^{2}$$
89 $$1 + 10T + 89T^{2}$$
97 $$1 - 16iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$