# Properties

 Label 2-3920-140.83-c0-0-0 Degree $2$ Conductor $3920$ Sign $-0.865 - 0.501i$ Analytic cond. $1.95633$ Root an. cond. $1.39869$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.382 + 0.923i)5-s − i·9-s + (−1.30 + 1.30i)13-s + (−0.541 − 0.541i)17-s + (−0.707 − 0.707i)25-s + 1.41i·29-s + (−1.41 + 1.41i)37-s + 1.84i·41-s + (0.923 + 0.382i)45-s + (−1 − i)53-s − 0.765i·61-s + (−0.707 − 1.70i)65-s + (−0.541 + 0.541i)73-s − 81-s + (0.707 − 0.292i)85-s + ⋯
 L(s)  = 1 + (−0.382 + 0.923i)5-s − i·9-s + (−1.30 + 1.30i)13-s + (−0.541 − 0.541i)17-s + (−0.707 − 0.707i)25-s + 1.41i·29-s + (−1.41 + 1.41i)37-s + 1.84i·41-s + (0.923 + 0.382i)45-s + (−1 − i)53-s − 0.765i·61-s + (−0.707 − 1.70i)65-s + (−0.541 + 0.541i)73-s − 81-s + (0.707 − 0.292i)85-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3920$$    =    $$2^{4} \cdot 5 \cdot 7^{2}$$ Sign: $-0.865 - 0.501i$ Analytic conductor: $$1.95633$$ Root analytic conductor: $$1.39869$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3920} (783, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3920,\ (\ :0),\ -0.865 - 0.501i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.4858231709$$ $$L(\frac12)$$ $$\approx$$ $$0.4858231709$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + (0.382 - 0.923i)T$$
7 $$1$$
good3 $$1 + iT^{2}$$
11 $$1 - T^{2}$$
13 $$1 + (1.30 - 1.30i)T - iT^{2}$$
17 $$1 + (0.541 + 0.541i)T + iT^{2}$$
19 $$1 - T^{2}$$
23 $$1 - iT^{2}$$
29 $$1 - 1.41iT - T^{2}$$
31 $$1 + T^{2}$$
37 $$1 + (1.41 - 1.41i)T - iT^{2}$$
41 $$1 - 1.84iT - T^{2}$$
43 $$1 - iT^{2}$$
47 $$1 - iT^{2}$$
53 $$1 + (1 + i)T + iT^{2}$$
59 $$1 - T^{2}$$
61 $$1 + 0.765iT - T^{2}$$
67 $$1 + iT^{2}$$
71 $$1 - T^{2}$$
73 $$1 + (0.541 - 0.541i)T - iT^{2}$$
79 $$1 + T^{2}$$
83 $$1 + iT^{2}$$
89 $$1 + 0.765T + T^{2}$$
97 $$1 + (1.30 + 1.30i)T + iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$