Properties

 Label 2-280-280.117-c1-0-33 Degree $2$ Conductor $280$ Sign $0.730 + 0.683i$ Analytic cond. $2.23581$ Root an. cond. $1.49526$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (1 − i)2-s + (1.86 + 0.5i)3-s − 2i·4-s + (−0.133 + 2.23i)5-s + (2.36 − 1.36i)6-s + (0.866 − 2.5i)7-s + (−2 − 2i)8-s + (0.633 + 0.366i)9-s + (2.09 + 2.36i)10-s + (0.633 − 0.366i)11-s + (1 − 3.73i)12-s + (−3 + 3i)13-s + (−1.63 − 3.36i)14-s + (−1.36 + 4.09i)15-s − 4·16-s + (4.09 + 1.09i)17-s + ⋯
 L(s)  = 1 + (0.707 − 0.707i)2-s + (1.07 + 0.288i)3-s − i·4-s + (−0.0599 + 0.998i)5-s + (0.965 − 0.557i)6-s + (0.327 − 0.944i)7-s + (−0.707 − 0.707i)8-s + (0.211 + 0.122i)9-s + (0.663 + 0.748i)10-s + (0.191 − 0.110i)11-s + (0.288 − 1.07i)12-s + (−0.832 + 0.832i)13-s + (−0.436 − 0.899i)14-s + (−0.352 + 1.05i)15-s − 16-s + (0.993 + 0.266i)17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$280$$    =    $$2^{3} \cdot 5 \cdot 7$$ Sign: $0.730 + 0.683i$ Analytic conductor: $$2.23581$$ Root analytic conductor: $$1.49526$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{280} (117, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 280,\ (\ :1/2),\ 0.730 + 0.683i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$2.20886 - 0.872017i$$ $$L(\frac12)$$ $$\approx$$ $$2.20886 - 0.872017i$$ $$L(\frac{3}{2})$$ 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 + (0.133 - 2.23i)T$$
7 $$1 + (-0.866 + 2.5i)T$$
good3 $$1 + (-1.86 - 0.5i)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 + (-4.09 - 1.09i)T + (14.7 + 8.5i)T^{2}$$
19 $$1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + (-0.401 - 1.5i)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 + (1.73 + 6.46i)T + (-32.0 + 18.5i)T^{2}$$
41 $$1 + 0.464iT - 41T^{2}$$
43 $$1 + (-2.36 - 2.36i)T + 43iT^{2}$$
47 $$1 + (-0.169 - 0.633i)T + (-40.7 + 23.5i)T^{2}$$
53 $$1 + (0.803 - 3i)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 + (3.86 + 1.03i)T + (58.0 + 33.5i)T^{2}$$
71 $$1 - 15.4T + 71T^{2}$$
73 $$1 + (2.16 - 8.09i)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}$$