# Properties

 Label 2-280-56.27-c1-0-24 Degree $2$ Conductor $280$ Sign $0.594 + 0.803i$ 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 + (1.41 − 0.0765i)2-s − 2.21i·3-s + (1.98 − 0.216i)4-s + 5-s + (−0.169 − 3.13i)6-s + (1.20 + 2.35i)7-s + (2.79 − 0.457i)8-s − 1.92·9-s + (1.41 − 0.0765i)10-s − 3.88·11-s + (−0.479 − 4.41i)12-s − 5.67·13-s + (1.88 + 3.23i)14-s − 2.21i·15-s + (3.90 − 0.859i)16-s + 5.63i·17-s + ⋯
 L(s)  = 1 + (0.998 − 0.0541i)2-s − 1.28i·3-s + (0.994 − 0.108i)4-s + 0.447·5-s + (−0.0693 − 1.27i)6-s + (0.457 + 0.889i)7-s + (0.986 − 0.161i)8-s − 0.641·9-s + (0.446 − 0.0242i)10-s − 1.17·11-s + (−0.138 − 1.27i)12-s − 1.57·13-s + (0.504 + 0.863i)14-s − 0.572i·15-s + (0.976 − 0.214i)16-s + 1.36i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$280$$    =    $$2^{3} \cdot 5 \cdot 7$$ Sign: $0.594 + 0.803i$ Analytic conductor: $$2.23581$$ Root analytic conductor: $$1.49526$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{280} (251, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 280,\ (\ :1/2),\ 0.594 + 0.803i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.12208 - 1.06955i$$ $$L(\frac12)$$ $$\approx$$ $$2.12208 - 1.06955i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1.41 + 0.0765i)T$$
5 $$1 - T$$
7 $$1 + (-1.20 - 2.35i)T$$
good3 $$1 + 2.21iT - 3T^{2}$$
11 $$1 + 3.88T + 11T^{2}$$
13 $$1 + 5.67T + 13T^{2}$$
17 $$1 - 5.63iT - 17T^{2}$$
19 $$1 + 1.31iT - 19T^{2}$$
23 $$1 + 7.37iT - 23T^{2}$$
29 $$1 - 9.07iT - 29T^{2}$$
31 $$1 + 2.23T + 31T^{2}$$
37 $$1 + 6.98iT - 37T^{2}$$
41 $$1 - 7.47iT - 41T^{2}$$
43 $$1 + 1.46T + 43T^{2}$$
47 $$1 - 0.567T + 47T^{2}$$
53 $$1 + 0.100iT - 53T^{2}$$
59 $$1 - 2.93iT - 59T^{2}$$
61 $$1 - 13.8T + 61T^{2}$$
67 $$1 - 5.54T + 67T^{2}$$
71 $$1 + 2.42iT - 71T^{2}$$
73 $$1 + 6.08iT - 73T^{2}$$
79 $$1 - 2.83iT - 79T^{2}$$
83 $$1 + 2.52iT - 83T^{2}$$
89 $$1 + 10.9iT - 89T^{2}$$
97 $$1 + 9.93iT - 97T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−12.30630825021736779579149156800, −11.04055124832184339465729467047, −10.13235382135660641724714500712, −8.493750873393390951845646909898, −7.57053726493223921267110337755, −6.67619074279668021922556507538, −5.65092994045240056348518486826, −4.78889198523495903299502945777, −2.67434687563554975762415297729, −1.94429898334409791617361001881, 2.49499067021003374299123461651, 3.83009246659029200246039803076, 4.96686673966736616266000417578, 5.31788223786157776101850468542, 7.09622336994217823571604007503, 7.83908950471242374627021749504, 9.744234682001624285425815499607, 10.05442543096129718343942039587, 11.07544895988698208386266164148, 11.88210931123598165614829184059