# Properties

 Label 2-1520-5.4-c1-0-27 Degree $2$ Conductor $1520$ Sign $1$ Analytic cond. $12.1372$ Root an. cond. $3.48385$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 0.874i·3-s + 2.23·5-s + 2.82i·7-s + 2.23·9-s + 0.763·11-s − 5.45i·13-s − 1.95i·15-s + 7.40i·17-s − 19-s + 2.47·21-s + 1.08i·23-s + 5.00·25-s − 4.57i·27-s + 4.47·29-s + 4·31-s + ⋯
 L(s)  = 1 − 0.504i·3-s + 0.999·5-s + 1.06i·7-s + 0.745·9-s + 0.230·11-s − 1.51i·13-s − 0.504i·15-s + 1.79i·17-s − 0.229·19-s + 0.539·21-s + 0.225i·23-s + 1.00·25-s − 0.880i·27-s + 0.830·29-s + 0.718·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1520$$    =    $$2^{4} \cdot 5 \cdot 19$$ Sign: $1$ Analytic conductor: $$12.1372$$ Root analytic conductor: $$3.48385$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1520} (609, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1520,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.256127227$$ $$L(\frac12)$$ $$\approx$$ $$2.256127227$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 - 2.23T$$
19 $$1 + T$$
good3 $$1 + 0.874iT - 3T^{2}$$
7 $$1 - 2.82iT - 7T^{2}$$
11 $$1 - 0.763T + 11T^{2}$$
13 $$1 + 5.45iT - 13T^{2}$$
17 $$1 - 7.40iT - 17T^{2}$$
23 $$1 - 1.08iT - 23T^{2}$$
29 $$1 - 4.47T + 29T^{2}$$
31 $$1 - 4T + 31T^{2}$$
37 $$1 - 2.62iT - 37T^{2}$$
41 $$1 + 6T + 41T^{2}$$
43 $$1 + 8.48iT - 43T^{2}$$
47 $$1 - 8.48iT - 47T^{2}$$
53 $$1 - 2.62iT - 53T^{2}$$
59 $$1 + 1.52T + 59T^{2}$$
61 $$1 + 11.7T + 61T^{2}$$
67 $$1 - 11.1iT - 67T^{2}$$
71 $$1 - 10.4T + 71T^{2}$$
73 $$1 - 5.24iT - 73T^{2}$$
79 $$1 - 15.4T + 79T^{2}$$
83 $$1 + 13.7iT - 83T^{2}$$
89 $$1 + 2.94T + 89T^{2}$$
97 $$1 + 13.9iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$