# Properties

 Label 2-520-104.101-c1-0-12 Degree $2$ Conductor $520$ Sign $0.798 - 0.602i$ Analytic cond. $4.15222$ Root an. cond. $2.03769$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.936 − 1.05i)2-s + (0.700 − 0.404i)3-s + (−0.245 + 1.98i)4-s − 5-s + (−1.08 − 0.363i)6-s + (3.38 + 1.95i)7-s + (2.33 − 1.59i)8-s + (−1.17 + 2.03i)9-s + (0.936 + 1.05i)10-s + (−0.223 − 0.387i)11-s + (0.630 + 1.48i)12-s + (−3.01 + 1.98i)13-s + (−1.09 − 5.41i)14-s + (−0.700 + 0.404i)15-s + (−3.87 − 0.975i)16-s + (−4.00 + 6.94i)17-s + ⋯
 L(s)  = 1 + (−0.662 − 0.749i)2-s + (0.404 − 0.233i)3-s + (−0.122 + 0.992i)4-s − 0.447·5-s + (−0.442 − 0.148i)6-s + (1.27 + 0.737i)7-s + (0.824 − 0.565i)8-s + (−0.390 + 0.677i)9-s + (0.296 + 0.335i)10-s + (−0.0674 − 0.116i)11-s + (0.182 + 0.429i)12-s + (−0.835 + 0.550i)13-s + (−0.293 − 1.44i)14-s + (−0.180 + 0.104i)15-s + (−0.969 − 0.243i)16-s + (−0.972 + 1.68i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$520$$    =    $$2^{3} \cdot 5 \cdot 13$$ Sign: $0.798 - 0.602i$ Analytic conductor: $$4.15222$$ Root analytic conductor: $$2.03769$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{520} (101, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 520,\ (\ :1/2),\ 0.798 - 0.602i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.929456 + 0.311530i$$ $$L(\frac12)$$ $$\approx$$ $$0.929456 + 0.311530i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.936 + 1.05i)T$$
5 $$1 + T$$
13 $$1 + (3.01 - 1.98i)T$$
good3 $$1 + (-0.700 + 0.404i)T + (1.5 - 2.59i)T^{2}$$
7 $$1 + (-3.38 - 1.95i)T + (3.5 + 6.06i)T^{2}$$
11 $$1 + (0.223 + 0.387i)T + (-5.5 + 9.52i)T^{2}$$
17 $$1 + (4.00 - 6.94i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (0.861 - 1.49i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (0.570 + 0.987i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-8.04 + 4.64i)T + (14.5 - 25.1i)T^{2}$$
31 $$1 - 5.19iT - 31T^{2}$$
37 $$1 + (-2.59 - 4.49i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (-4.37 + 2.52i)T + (20.5 - 35.5i)T^{2}$$
43 $$1 + (-0.587 - 0.339i)T + (21.5 + 37.2i)T^{2}$$
47 $$1 + 8.71iT - 47T^{2}$$
53 $$1 + 11.8iT - 53T^{2}$$
59 $$1 + (4.67 - 8.08i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (6.84 + 3.95i)T + (30.5 + 52.8i)T^{2}$$
67 $$1 + (-5.35 - 9.28i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + (-9.05 - 5.22i)T + (35.5 + 61.4i)T^{2}$$
73 $$1 + 0.118iT - 73T^{2}$$
79 $$1 - 17.2T + 79T^{2}$$
83 $$1 + 10.1T + 83T^{2}$$
89 $$1 + (-7.70 + 4.44i)T + (44.5 - 77.0i)T^{2}$$
97 $$1 + (-7.84 - 4.52i)T + (48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$