# Properties

 Label 2-240-5.4-c3-0-6 Degree $2$ Conductor $240$ Sign $0.447 - 0.894i$ Analytic cond. $14.1604$ Root an. cond. $3.76303$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3i·3-s + (−10 − 5i)5-s − 10i·7-s − 9·9-s + 46·11-s + 34i·13-s + (15 − 30i)15-s + 66i·17-s + 104·19-s + 30·21-s + 164i·23-s + (75 + 100i)25-s − 27i·27-s − 224·29-s + 72·31-s + ⋯
 L(s)  = 1 + 0.577i·3-s + (−0.894 − 0.447i)5-s − 0.539i·7-s − 0.333·9-s + 1.26·11-s + 0.725i·13-s + (0.258 − 0.516i)15-s + 0.941i·17-s + 1.25·19-s + 0.311·21-s + 1.48i·23-s + (0.599 + 0.800i)25-s − 0.192i·27-s − 1.43·29-s + 0.417·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$240$$    =    $$2^{4} \cdot 3 \cdot 5$$ Sign: $0.447 - 0.894i$ Analytic conductor: $$14.1604$$ Root analytic conductor: $$3.76303$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{240} (49, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 240,\ (\ :3/2),\ 0.447 - 0.894i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.23325 + 0.762193i$$ $$L(\frac12)$$ $$\approx$$ $$1.23325 + 0.762193i$$ $$L(\frac{5}{2})$$ 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 - 3iT$$
5 $$1 + (10 + 5i)T$$
good7 $$1 + 10iT - 343T^{2}$$
11 $$1 - 46T + 1.33e3T^{2}$$
13 $$1 - 34iT - 2.19e3T^{2}$$
17 $$1 - 66iT - 4.91e3T^{2}$$
19 $$1 - 104T + 6.85e3T^{2}$$
23 $$1 - 164iT - 1.21e4T^{2}$$
29 $$1 + 224T + 2.43e4T^{2}$$
31 $$1 - 72T + 2.97e4T^{2}$$
37 $$1 + 22iT - 5.06e4T^{2}$$
41 $$1 - 194T + 6.89e4T^{2}$$
43 $$1 - 108iT - 7.95e4T^{2}$$
47 $$1 - 480iT - 1.03e5T^{2}$$
53 $$1 + 286iT - 1.48e5T^{2}$$
59 $$1 - 426T + 2.05e5T^{2}$$
61 $$1 - 698T + 2.26e5T^{2}$$
67 $$1 + 328iT - 3.00e5T^{2}$$
71 $$1 + 188T + 3.57e5T^{2}$$
73 $$1 - 740iT - 3.89e5T^{2}$$
79 $$1 - 1.16e3T + 4.93e5T^{2}$$
83 $$1 - 412iT - 5.71e5T^{2}$$
89 $$1 + 1.20e3T + 7.04e5T^{2}$$
97 $$1 + 1.38e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$