# Properties

 Label 2-280-280.173-c1-0-26 Degree $2$ Conductor $280$ Sign $-0.937 + 0.347i$ Analytic cond. $2.23581$ Root an. cond. $1.49526$ 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 + (−0.5 + 1.86i)3-s + 2i·4-s + (−1.86 + 1.23i)5-s + (2.36 − 1.36i)6-s + (−2.5 − 0.866i)7-s + (2 − 2i)8-s + (−0.633 − 0.366i)9-s + (3.09 + 0.633i)10-s + (0.633 − 0.366i)11-s + (−3.73 − i)12-s + (−3 − 3i)13-s + (1.63 + 3.36i)14-s + (−1.36 − 4.09i)15-s − 4·16-s + (1.09 − 4.09i)17-s + ⋯
 L(s)  = 1 + (−0.707 − 0.707i)2-s + (−0.288 + 1.07i)3-s + i·4-s + (−0.834 + 0.550i)5-s + (0.965 − 0.557i)6-s + (−0.944 − 0.327i)7-s + (0.707 − 0.707i)8-s + (−0.211 − 0.122i)9-s + (0.979 + 0.200i)10-s + (0.191 − 0.110i)11-s + (−1.07 − 0.288i)12-s + (−0.832 − 0.832i)13-s + (0.436 + 0.899i)14-s + (−0.352 − 1.05i)15-s − 16-s + (0.266 − 0.993i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$280$$    =    $$2^{3} \cdot 5 \cdot 7$$ Sign: $-0.937 + 0.347i$ Analytic conductor: $$2.23581$$ Root analytic conductor: $$1.49526$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{280} (173, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$1$$ Selberg data: $$(2,\ 280,\ (\ :1/2),\ -0.937 + 0.347i)$$

## 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$$F_p(T)$
bad2 $$1 + (1 + i)T$$
5 $$1 + (1.86 - 1.23i)T$$
7 $$1 + (2.5 + 0.866i)T$$
good3 $$1 + (0.5 - 1.86i)T + (-2.59 - 1.5i)T^{2}$$
11 $$1 + (-0.633 + 0.366i)T + (5.5 - 9.52i)T^{2}$$
13 $$1 + (3 + 3i)T + 13iT^{2}$$
17 $$1 + (-1.09 + 4.09i)T + (-14.7 - 8.5i)T^{2}$$
19 $$1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + (1.5 - 0.401i)T + (19.9 - 11.5i)T^{2}$$
29 $$1 - 4.46T + 29T^{2}$$
31 $$1 + (8.83 - 5.09i)T + (15.5 - 26.8i)T^{2}$$
37 $$1 + (6.46 - 1.73i)T + (32.0 - 18.5i)T^{2}$$
41 $$1 + 0.464iT - 41T^{2}$$
43 $$1 + (2.36 - 2.36i)T - 43iT^{2}$$
47 $$1 + (-0.633 + 0.169i)T + (40.7 - 23.5i)T^{2}$$
53 $$1 + (3 + 0.803i)T + (45.8 + 26.5i)T^{2}$$
59 $$1 + (7.73 - 4.46i)T + (29.5 - 51.0i)T^{2}$$
61 $$1 + (-4.96 + 8.59i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (1.03 - 3.86i)T + (-58.0 - 33.5i)T^{2}$$
71 $$1 - 15.4T + 71T^{2}$$
73 $$1 + (8.09 + 2.16i)T + (63.2 + 36.5i)T^{2}$$
79 $$1 + (11.1 + 6.46i)T + (39.5 + 68.4i)T^{2}$$
83 $$1 + (4.56 + 4.56i)T + 83iT^{2}$$
89 $$1 + (8.59 - 14.8i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (-6.26 - 6.26i)T + 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$