# Properties

 Label 2-285-15.2-c1-0-26 Degree $2$ Conductor $285$ Sign $-0.880 + 0.473i$ Analytic cond. $2.27573$ Root an. cond. $1.50855$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 + i)2-s − 1.73i·3-s + (−0.133 + 2.23i)5-s + (1.73 + 1.73i)6-s + (−2.36 − 2.36i)7-s + (−2 − 2i)8-s − 2.99·9-s + (−2.09 − 2.36i)10-s − 0.267i·11-s + (−3.73 + 3.73i)13-s + 4.73·14-s + (3.86 + 0.232i)15-s + 4·16-s + (0.0980 − 0.0980i)17-s + (2.99 − 2.99i)18-s − i·19-s + ⋯
 L(s)  = 1 + (−0.707 + 0.707i)2-s − 0.999i·3-s + (−0.0599 + 0.998i)5-s + (0.707 + 0.707i)6-s + (−0.894 − 0.894i)7-s + (−0.707 − 0.707i)8-s − 0.999·9-s + (−0.663 − 0.748i)10-s − 0.0807i·11-s + (−1.03 + 1.03i)13-s + 1.26·14-s + (0.998 + 0.0599i)15-s + 16-s + (0.0237 − 0.0237i)17-s + (0.707 − 0.707i)18-s − 0.229i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$285$$    =    $$3 \cdot 5 \cdot 19$$ Sign: $-0.880 + 0.473i$ Analytic conductor: $$2.27573$$ Root analytic conductor: $$1.50855$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{285} (77, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$1$$ Selberg data: $$(2,\ 285,\ (\ :1/2),\ -0.880 + 0.473i)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + 1.73iT$$
5 $$1 + (0.133 - 2.23i)T$$
19 $$1 + iT$$
good2 $$1 + (1 - i)T - 2iT^{2}$$
7 $$1 + (2.36 + 2.36i)T + 7iT^{2}$$
11 $$1 + 0.267iT - 11T^{2}$$
13 $$1 + (3.73 - 3.73i)T - 13iT^{2}$$
17 $$1 + (-0.0980 + 0.0980i)T - 17iT^{2}$$
23 $$1 + (6.46 + 6.46i)T + 23iT^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + 8.73T + 31T^{2}$$
37 $$1 + (-2.46 - 2.46i)T + 37iT^{2}$$
41 $$1 - 7.66iT - 41T^{2}$$
43 $$1 + (-0.0980 + 0.0980i)T - 43iT^{2}$$
47 $$1 + (-3.56 + 3.56i)T - 47iT^{2}$$
53 $$1 + (-2.26 - 2.26i)T + 53iT^{2}$$
59 $$1 - 10.1T + 59T^{2}$$
61 $$1 + 4.26T + 61T^{2}$$
67 $$1 + (-8.19 - 8.19i)T + 67iT^{2}$$
71 $$1 - 7.26iT - 71T^{2}$$
73 $$1 + (-3.63 + 3.63i)T - 73iT^{2}$$
79 $$1 + 10.3iT - 79T^{2}$$
83 $$1 + (3.73 + 3.73i)T + 83iT^{2}$$
89 $$1 + 10.7T + 89T^{2}$$
97 $$1 + (-0.464 - 0.464i)T + 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$