# Properties

 Label 2-280-7.4-c1-0-4 Degree $2$ Conductor $280$ Sign $0.952 + 0.305i$ 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 + (−0.352 + 0.611i)3-s + (−0.5 − 0.866i)5-s + (0.647 − 2.56i)7-s + (1.25 + 2.16i)9-s + (2.25 − 3.89i)11-s + 5.09·13-s + 0.705·15-s + (−1 + 1.73i)17-s + (1.54 + 2.67i)19-s + (1.33 + 1.30i)21-s + (−2.89 − 5.01i)23-s + (−0.499 + 0.866i)25-s − 3.88·27-s + 9.50·29-s + (−2.70 + 4.68i)31-s + ⋯
 L(s)  = 1 + (−0.203 + 0.352i)3-s + (−0.223 − 0.387i)5-s + (0.244 − 0.969i)7-s + (0.416 + 0.722i)9-s + (0.678 − 1.17i)11-s + 1.41·13-s + 0.182·15-s + (−0.242 + 0.420i)17-s + (0.354 + 0.614i)19-s + (0.292 + 0.283i)21-s + (−0.604 − 1.04i)23-s + (−0.0999 + 0.173i)25-s − 0.747·27-s + 1.76·29-s + (−0.485 + 0.841i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$280$$    =    $$2^{3} \cdot 5 \cdot 7$$ Sign: $0.952 + 0.305i$ Analytic conductor: $$2.23581$$ Root analytic conductor: $$1.49526$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{280} (81, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 280,\ (\ :1/2),\ 0.952 + 0.305i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.26666 - 0.198230i$$ $$L(\frac12)$$ $$\approx$$ $$1.26666 - 0.198230i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + (0.5 + 0.866i)T$$
7 $$1 + (-0.647 + 2.56i)T$$
good3 $$1 + (0.352 - 0.611i)T + (-1.5 - 2.59i)T^{2}$$
11 $$1 + (-2.25 + 3.89i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 - 5.09T + 13T^{2}$$
17 $$1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-1.54 - 2.67i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (2.89 + 5.01i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 - 9.50T + 29T^{2}$$
31 $$1 + (2.70 - 4.68i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (3.54 + 6.14i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + 6.59T + 41T^{2}$$
43 $$1 - 4.70T + 43T^{2}$$
47 $$1 + (5.04 + 8.74i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (4.95 - 8.58i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-4.45 - 7.71i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-0.0585 + 0.101i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + (4.09 - 7.08i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (-7.09 - 12.2i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + 10.7T + 83T^{2}$$
89 $$1 + (2.04 + 3.54i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + 2T + 97T^{2}$$
show less
$$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

−11.63274242187820743390002973300, −10.74445708067111689060542361036, −10.27329657001604549897501625188, −8.718289668223804175196991506986, −8.184455428919357730852454491593, −6.84508144592908127245152150641, −5.75181646687244594181674559072, −4.41664561998213504208498772708, −3.62295099703294338690351473884, −1.26543248979319791345994143738, 1.65957801706493661487406782341, 3.37664838505248924435401190206, 4.71358651196764302376313780721, 6.16521403520588153953849409205, 6.81705453049961231551908910811, 8.011370545677994847968891202965, 9.131758329171966546045165460147, 9.874853288208778746271831505979, 11.31849868875329482345731270712, 11.82270515582679783150880397177