# Properties

 Label 2-3360-840.797-c0-0-0 Degree $2$ Conductor $3360$ Sign $0.850 - 0.525i$ Analytic cond. $1.67685$ Root an. cond. $1.29493$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.923 − 0.382i)3-s + (−0.382 − 0.923i)5-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)9-s + (−1.30 + 1.30i)13-s + i·15-s + 0.765i·19-s + (0.382 + 0.923i)21-s + (1 + i)23-s + (−0.707 + 0.707i)25-s + (−0.382 − 0.923i)27-s + (−0.382 + 0.923i)35-s + (1.70 − 0.707i)39-s + (0.382 − 0.923i)45-s + 1.00i·49-s + ⋯
 L(s)  = 1 + (−0.923 − 0.382i)3-s + (−0.382 − 0.923i)5-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)9-s + (−1.30 + 1.30i)13-s + i·15-s + 0.765i·19-s + (0.382 + 0.923i)21-s + (1 + i)23-s + (−0.707 + 0.707i)25-s + (−0.382 − 0.923i)27-s + (−0.382 + 0.923i)35-s + (1.70 − 0.707i)39-s + (0.382 − 0.923i)45-s + 1.00i·49-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3360$$    =    $$2^{5} \cdot 3 \cdot 5 \cdot 7$$ Sign: $0.850 - 0.525i$ Analytic conductor: $$1.67685$$ Root analytic conductor: $$1.29493$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3360} (2897, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3360,\ (\ :0),\ 0.850 - 0.525i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5217809761$$ $$L(\frac12)$$ $$\approx$$ $$0.5217809761$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (0.923 + 0.382i)T$$
5 $$1 + (0.382 + 0.923i)T$$
7 $$1 + (0.707 + 0.707i)T$$
good11 $$1 + T^{2}$$
13 $$1 + (1.30 - 1.30i)T - iT^{2}$$
17 $$1 + iT^{2}$$
19 $$1 - 0.765iT - T^{2}$$
23 $$1 + (-1 - i)T + iT^{2}$$
29 $$1 - T^{2}$$
31 $$1 - T^{2}$$
37 $$1 - iT^{2}$$
41 $$1 + T^{2}$$
43 $$1 + iT^{2}$$
47 $$1 + iT^{2}$$
53 $$1 - iT^{2}$$
59 $$1 - 0.765T + T^{2}$$
61 $$1 - 1.84T + T^{2}$$
67 $$1 - iT^{2}$$
71 $$1 + 1.41iT - T^{2}$$
73 $$1 + iT^{2}$$
79 $$1 - 1.41iT - T^{2}$$
83 $$1 + (0.541 + 0.541i)T + iT^{2}$$
89 $$1 - T^{2}$$
97 $$1 - iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$