# Properties

 Label 2-3840-8.5-c1-0-4 Degree $2$ Conductor $3840$ Sign $-0.707 - 0.707i$ Analytic cond. $30.6625$ Root an. cond. $5.53737$ Motivic weight $1$ 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 − 9-s + 4i·11-s + 2i·13-s + 15-s + 2·17-s + 4i·19-s − 25-s + i·27-s + 2i·29-s + 4·33-s − 10i·37-s + 2·39-s − 10·41-s + ⋯
 L(s)  = 1 − 0.577i·3-s + 0.447i·5-s − 0.333·9-s + 1.20i·11-s + 0.554i·13-s + 0.258·15-s + 0.485·17-s + 0.917i·19-s − 0.200·25-s + 0.192i·27-s + 0.371i·29-s + 0.696·33-s − 1.64i·37-s + 0.320·39-s − 1.56·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3840$$    =    $$2^{8} \cdot 3 \cdot 5$$ Sign: $-0.707 - 0.707i$ Analytic conductor: $$30.6625$$ Root analytic conductor: $$5.53737$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{3840} (1921, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3840,\ (\ :1/2),\ -0.707 - 0.707i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.7981211110$$ $$L(\frac12)$$ $$\approx$$ $$0.7981211110$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + iT$$
5 $$1 - iT$$
good7 $$1 + 7T^{2}$$
11 $$1 - 4iT - 11T^{2}$$
13 $$1 - 2iT - 13T^{2}$$
17 $$1 - 2T + 17T^{2}$$
19 $$1 - 4iT - 19T^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 - 2iT - 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 + 10iT - 37T^{2}$$
41 $$1 + 10T + 41T^{2}$$
43 $$1 + 4iT - 43T^{2}$$
47 $$1 + 8T + 47T^{2}$$
53 $$1 + 10iT - 53T^{2}$$
59 $$1 - 4iT - 59T^{2}$$
61 $$1 - 2iT - 61T^{2}$$
67 $$1 - 12iT - 67T^{2}$$
71 $$1 + 8T + 71T^{2}$$
73 $$1 + 10T + 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 - 12iT - 83T^{2}$$
89 $$1 - 6T + 89T^{2}$$
97 $$1 - 2T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$