# Properties

 Label 2-3360-12.11-c1-0-0 Degree $2$ Conductor $3360$ Sign $-0.291 + 0.956i$ Analytic cond. $26.8297$ Root an. cond. $5.17974$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.814 + 1.52i)3-s + i·5-s + i·7-s + (−1.67 + 2.49i)9-s − 1.02·11-s − 4.05·13-s + (−1.52 + 0.814i)15-s − 3.17i·17-s + 2.43i·19-s + (−1.52 + 0.814i)21-s − 6.89·23-s − 25-s + (−5.16 − 0.527i)27-s − 4.16i·29-s − 2.76i·31-s + ⋯
 L(s)  = 1 + (0.470 + 0.882i)3-s + 0.447i·5-s + 0.377i·7-s + (−0.557 + 0.830i)9-s − 0.310·11-s − 1.12·13-s + (−0.394 + 0.210i)15-s − 0.770i·17-s + 0.558i·19-s + (−0.333 + 0.177i)21-s − 1.43·23-s − 0.200·25-s + (−0.994 − 0.101i)27-s − 0.772i·29-s − 0.496i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3360$$    =    $$2^{5} \cdot 3 \cdot 5 \cdot 7$$ Sign: $-0.291 + 0.956i$ Analytic conductor: $$26.8297$$ Root analytic conductor: $$5.17974$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{3360} (2591, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3360,\ (\ :1/2),\ -0.291 + 0.956i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.01267571384$$ $$L(\frac12)$$ $$\approx$$ $$0.01267571384$$ $$L(\frac{3}{2})$$ 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 + (-0.814 - 1.52i)T$$
5 $$1 - iT$$
7 $$1 - iT$$
good11 $$1 + 1.02T + 11T^{2}$$
13 $$1 + 4.05T + 13T^{2}$$
17 $$1 + 3.17iT - 17T^{2}$$
19 $$1 - 2.43iT - 19T^{2}$$
23 $$1 + 6.89T + 23T^{2}$$
29 $$1 + 4.16iT - 29T^{2}$$
31 $$1 + 2.76iT - 31T^{2}$$
37 $$1 - 0.949T + 37T^{2}$$
41 $$1 + 5.27iT - 41T^{2}$$
43 $$1 + 9.82iT - 43T^{2}$$
47 $$1 - 5.55T + 47T^{2}$$
53 $$1 - 2.52iT - 53T^{2}$$
59 $$1 - 13.4T + 59T^{2}$$
61 $$1 + 13.8T + 61T^{2}$$
67 $$1 - 12.3iT - 67T^{2}$$
71 $$1 - 0.483T + 71T^{2}$$
73 $$1 + 2.55T + 73T^{2}$$
79 $$1 - 4.87iT - 79T^{2}$$
83 $$1 + 8.24T + 83T^{2}$$
89 $$1 + 2.15iT - 89T^{2}$$
97 $$1 - 1.82T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$