# Properties

 Label 2-63e2-3.2-c0-0-2 Degree $2$ Conductor $3969$ Sign $-1$ Analytic cond. $1.98078$ Root an. cond. $1.40740$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.93i·2-s − 2.73·4-s + 3.34i·8-s + 0.517i·11-s + 3.73·16-s + 0.999·22-s − 1.41i·23-s + 25-s − 1.41i·29-s − 3.86i·32-s + 1.73·37-s − 1.73·43-s − 1.41i·44-s − 2.73·46-s − 1.93i·50-s + ⋯
 L(s)  = 1 − 1.93i·2-s − 2.73·4-s + 3.34i·8-s + 0.517i·11-s + 3.73·16-s + 0.999·22-s − 1.41i·23-s + 25-s − 1.41i·29-s − 3.86i·32-s + 1.73·37-s − 1.73·43-s − 1.41i·44-s − 2.73·46-s − 1.93i·50-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3969$$    =    $$3^{4} \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$1.98078$$ Root analytic conductor: $$1.40740$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3969} (3725, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3969,\ (\ :0),\ -1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.9553796115$$ $$L(\frac12)$$ $$\approx$$ $$0.9553796115$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1$$
good2 $$1 + 1.93iT - T^{2}$$
5 $$1 - T^{2}$$
11 $$1 - 0.517iT - T^{2}$$
13 $$1 + T^{2}$$
17 $$1 - T^{2}$$
19 $$1 + T^{2}$$
23 $$1 + 1.41iT - T^{2}$$
29 $$1 + 1.41iT - T^{2}$$
31 $$1 + T^{2}$$
37 $$1 - 1.73T + T^{2}$$
41 $$1 - T^{2}$$
43 $$1 + 1.73T + T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + 0.517iT - T^{2}$$
59 $$1 - T^{2}$$
61 $$1 + T^{2}$$
67 $$1 - T + T^{2}$$
71 $$1 + 1.93iT - T^{2}$$
73 $$1 + T^{2}$$
79 $$1 + T + 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}$$