# Properties

 Label 2-63e2-63.23-c0-0-5 Degree $2$ Conductor $3969$ Sign $-0.458 + 0.888i$ 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 − 1.41i·2-s − 1.00·4-s + (1.22 + 0.707i)11-s − 0.999·16-s + (1.00 − 1.73i)22-s + (1.22 − 0.707i)23-s + (−0.5 + 0.866i)25-s + (1.22 − 0.707i)29-s + 1.41i·32-s + (−1.22 − 0.707i)44-s + (−1.00 − 1.73i)46-s + (1.22 + 0.707i)50-s + (1.22 − 0.707i)53-s + (−1.00 − 1.73i)58-s + 1.00·64-s + ⋯
 L(s)  = 1 − 1.41i·2-s − 1.00·4-s + (1.22 + 0.707i)11-s − 0.999·16-s + (1.00 − 1.73i)22-s + (1.22 − 0.707i)23-s + (−0.5 + 0.866i)25-s + (1.22 − 0.707i)29-s + 1.41i·32-s + (−1.22 − 0.707i)44-s + (−1.00 − 1.73i)46-s + (1.22 + 0.707i)50-s + (1.22 − 0.707i)53-s + (−1.00 − 1.73i)58-s + 1.00·64-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3969$$    =    $$3^{4} \cdot 7^{2}$$ Sign: $-0.458 + 0.888i$ Analytic conductor: $$1.98078$$ Root analytic conductor: $$1.40740$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3969} (863, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3969,\ (\ :0),\ -0.458 + 0.888i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.414282318$$ $$L(\frac12)$$ $$\approx$$ $$1.414282318$$ $$L(1)$$ 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 + 1.41iT - T^{2}$$
5 $$1 + (0.5 - 0.866i)T^{2}$$
11 $$1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2}$$
13 $$1 + (-0.5 - 0.866i)T^{2}$$
17 $$1 + (0.5 - 0.866i)T^{2}$$
19 $$1 + (-0.5 - 0.866i)T^{2}$$
23 $$1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2}$$
29 $$1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2}$$
31 $$1 + T^{2}$$
37 $$1 + (-0.5 - 0.866i)T^{2}$$
41 $$1 + (0.5 + 0.866i)T^{2}$$
43 $$1 + (-0.5 + 0.866i)T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2}$$
59 $$1 - T^{2}$$
61 $$1 + T^{2}$$
67 $$1 + 2T + T^{2}$$
71 $$1 + 1.41iT - T^{2}$$
73 $$1 + (-0.5 + 0.866i)T^{2}$$
79 $$1 - 2T + T^{2}$$
83 $$1 + (0.5 - 0.866i)T^{2}$$
89 $$1 + (0.5 + 0.866i)T^{2}$$
97 $$1 + (-0.5 + 0.866i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$