# Properties

 Label 2-63e2-441.11-c0-0-0 Degree $2$ Conductor $3969$ Sign $0.484 - 0.874i$ 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 + (−0.900 + 0.433i)4-s + (0.955 + 0.294i)7-s + (−0.535 + 0.496i)13-s + (0.623 − 0.781i)16-s + (0.222 + 0.385i)19-s + (0.955 − 0.294i)25-s + (−0.988 + 0.149i)28-s + 0.149·31-s + (0.0111 + 0.149i)37-s + (0.698 + 1.77i)43-s + (0.826 + 0.563i)49-s + (0.266 − 0.680i)52-s + (−1.48 − 0.716i)61-s + (−0.222 + 0.974i)64-s − 1.97·67-s + ⋯
 L(s)  = 1 + (−0.900 + 0.433i)4-s + (0.955 + 0.294i)7-s + (−0.535 + 0.496i)13-s + (0.623 − 0.781i)16-s + (0.222 + 0.385i)19-s + (0.955 − 0.294i)25-s + (−0.988 + 0.149i)28-s + 0.149·31-s + (0.0111 + 0.149i)37-s + (0.698 + 1.77i)43-s + (0.826 + 0.563i)49-s + (0.266 − 0.680i)52-s + (−1.48 − 0.716i)61-s + (−0.222 + 0.974i)64-s − 1.97·67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3969$$    =    $$3^{4} \cdot 7^{2}$$ Sign: $0.484 - 0.874i$ Analytic conductor: $$1.98078$$ Root analytic conductor: $$1.40740$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3969} (3833, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3969,\ (\ :0),\ 0.484 - 0.874i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.102838134$$ $$L(\frac12)$$ $$\approx$$ $$1.102838134$$ $$L(1)$$ 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 + (-0.955 - 0.294i)T$$
good2 $$1 + (0.900 - 0.433i)T^{2}$$
5 $$1 + (-0.955 + 0.294i)T^{2}$$
11 $$1 + (-0.0747 + 0.997i)T^{2}$$
13 $$1 + (0.535 - 0.496i)T + (0.0747 - 0.997i)T^{2}$$
17 $$1 + (-0.365 + 0.930i)T^{2}$$
19 $$1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2}$$
23 $$1 + (0.988 + 0.149i)T^{2}$$
29 $$1 + (-0.365 + 0.930i)T^{2}$$
31 $$1 - 0.149T + T^{2}$$
37 $$1 + (-0.0111 - 0.149i)T + (-0.988 + 0.149i)T^{2}$$
41 $$1 + (0.733 + 0.680i)T^{2}$$
43 $$1 + (-0.698 - 1.77i)T + (-0.733 + 0.680i)T^{2}$$
47 $$1 + (0.900 - 0.433i)T^{2}$$
53 $$1 + (0.988 + 0.149i)T^{2}$$
59 $$1 + (0.222 + 0.974i)T^{2}$$
61 $$1 + (1.48 + 0.716i)T + (0.623 + 0.781i)T^{2}$$
67 $$1 + 1.97T + T^{2}$$
71 $$1 + (-0.623 + 0.781i)T^{2}$$
73 $$1 + (-1.32 - 1.22i)T + (0.0747 + 0.997i)T^{2}$$
79 $$1 - 1.65T + T^{2}$$
83 $$1 + (-0.0747 - 0.997i)T^{2}$$
89 $$1 + (-0.826 + 0.563i)T^{2}$$
97 $$1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$