# Properties

 Label 2-3328-52.51-c0-0-12 Degree $2$ Conductor $3328$ Sign $-1$ Analytic cond. $1.66088$ Root an. cond. $1.28875$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·3-s − i·5-s − 7-s − i·13-s − 15-s − 17-s + i·21-s − i·27-s − 2·31-s + i·35-s − i·37-s − 39-s + i·43-s + 47-s + i·51-s + ⋯
 L(s)  = 1 − i·3-s − i·5-s − 7-s − i·13-s − 15-s − 17-s + i·21-s − i·27-s − 2·31-s + i·35-s − i·37-s − 39-s + i·43-s + 47-s + i·51-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3328$$    =    $$2^{8} \cdot 13$$ Sign: $-1$ Analytic conductor: $$1.66088$$ Root analytic conductor: $$1.28875$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3328} (3327, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3328,\ (\ :0),\ -1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.7906141331$$ $$L(\frac12)$$ $$\approx$$ $$0.7906141331$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
13 $$1 + iT$$
good3 $$1 + iT - T^{2}$$
5 $$1 + iT - T^{2}$$
7 $$1 + T + T^{2}$$
11 $$1 + T^{2}$$
17 $$1 + T + T^{2}$$
19 $$1 + T^{2}$$
23 $$1 - T^{2}$$
29 $$1 + T^{2}$$
31 $$1 + 2T + T^{2}$$
37 $$1 + iT - T^{2}$$
41 $$1 - T^{2}$$
43 $$1 - iT - T^{2}$$
47 $$1 - T + T^{2}$$
53 $$1 + T^{2}$$
59 $$1 + T^{2}$$
61 $$1 + T^{2}$$
67 $$1 + T^{2}$$
71 $$1 + T + T^{2}$$
73 $$1 - T^{2}$$
79 $$1 - T^{2}$$
83 $$1 + T^{2}$$
89 $$1 - T^{2}$$
97 $$1 - T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$