# Properties

 Label 2-840-5.4-c1-0-15 Degree $2$ Conductor $840$ Sign $0.749 + 0.662i$ Analytic cond. $6.70743$ Root an. cond. $2.58987$ 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 + (1.48 − 1.67i)5-s − i·7-s − 9-s + 0.387·11-s − 2.96i·13-s + (1.67 + 1.48i)15-s − 3.35i·17-s + 2.96·19-s + 21-s − 0.962i·23-s + (−0.612 − 4.96i)25-s − i·27-s − 1.22·29-s + 2.96·31-s + ⋯
 L(s)  = 1 + 0.577i·3-s + (0.662 − 0.749i)5-s − 0.377i·7-s − 0.333·9-s + 0.116·11-s − 0.821i·13-s + (0.432 + 0.382i)15-s − 0.812i·17-s + 0.679·19-s + 0.218·21-s − 0.200i·23-s + (−0.122 − 0.992i)25-s − 0.192i·27-s − 0.227·29-s + 0.532·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$840$$    =    $$2^{3} \cdot 3 \cdot 5 \cdot 7$$ Sign: $0.749 + 0.662i$ Analytic conductor: $$6.70743$$ Root analytic conductor: $$2.58987$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{840} (169, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 840,\ (\ :1/2),\ 0.749 + 0.662i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.56551 - 0.592870i$$ $$L(\frac12)$$ $$\approx$$ $$1.56551 - 0.592870i$$ $$L(\frac{3}{2})$$ 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 + (-1.48 + 1.67i)T$$
7 $$1 + iT$$
good11 $$1 - 0.387T + 11T^{2}$$
13 $$1 + 2.96iT - 13T^{2}$$
17 $$1 + 3.35iT - 17T^{2}$$
19 $$1 - 2.96T + 19T^{2}$$
23 $$1 + 0.962iT - 23T^{2}$$
29 $$1 + 1.22T + 29T^{2}$$
31 $$1 - 2.96T + 31T^{2}$$
37 $$1 + 5.92iT - 37T^{2}$$
41 $$1 - 1.03T + 41T^{2}$$
43 $$1 - 10.7iT - 43T^{2}$$
47 $$1 + 3.22iT - 47T^{2}$$
53 $$1 + 5.66iT - 53T^{2}$$
59 $$1 + 3.22T + 59T^{2}$$
61 $$1 - 14.6T + 61T^{2}$$
67 $$1 - 67T^{2}$$
71 $$1 - 5.53T + 71T^{2}$$
73 $$1 - 6.18iT - 73T^{2}$$
79 $$1 - 13.9T + 79T^{2}$$
83 $$1 - 3.22iT - 83T^{2}$$
89 $$1 + 3.73T + 89T^{2}$$
97 $$1 - 7.73iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$