# Properties

 Label 2-280-280.19-c1-0-20 Degree $2$ Conductor $280$ Sign $0.138 + 0.990i$ Analytic cond. $2.23581$ Root an. cond. $1.49526$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−1.39 + 0.230i)2-s + (−0.929 − 1.60i)3-s + (1.89 − 0.642i)4-s + (2.18 − 0.457i)5-s + (1.66 + 2.03i)6-s + (1.10 − 2.40i)7-s + (−2.49 + 1.33i)8-s + (−0.226 + 0.392i)9-s + (−2.94 + 1.14i)10-s + (1.61 + 2.78i)11-s + (−2.79 − 2.45i)12-s + 0.668i·13-s + (−0.989 + 3.60i)14-s + (−2.77 − 3.09i)15-s + (3.17 − 2.43i)16-s + (1.62 + 2.81i)17-s + ⋯
 L(s)  = 1 + (−0.986 + 0.162i)2-s + (−0.536 − 0.929i)3-s + (0.946 − 0.321i)4-s + (0.978 − 0.204i)5-s + (0.680 + 0.829i)6-s + (0.418 − 0.908i)7-s + (−0.881 + 0.471i)8-s + (−0.0755 + 0.130i)9-s + (−0.932 + 0.361i)10-s + (0.485 + 0.841i)11-s + (−0.806 − 0.707i)12-s + 0.185i·13-s + (−0.264 + 0.964i)14-s + (−0.715 − 0.799i)15-s + (0.793 − 0.608i)16-s + (0.393 + 0.681i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$280$$    =    $$2^{3} \cdot 5 \cdot 7$$ Sign: $0.138 + 0.990i$ Analytic conductor: $$2.23581$$ Root analytic conductor: $$1.49526$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{280} (19, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 280,\ (\ :1/2),\ 0.138 + 0.990i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.650442 - 0.565885i$$ $$L(\frac12)$$ $$\approx$$ $$0.650442 - 0.565885i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.39 - 0.230i)T$$
5 $$1 + (-2.18 + 0.457i)T$$
7 $$1 + (-1.10 + 2.40i)T$$
good3 $$1 + (0.929 + 1.60i)T + (-1.5 + 2.59i)T^{2}$$
11 $$1 + (-1.61 - 2.78i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 - 0.668iT - 13T^{2}$$
17 $$1 + (-1.62 - 2.81i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (5.77 + 3.33i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + (-4.41 + 7.64i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + 3.17iT - 29T^{2}$$
31 $$1 + (2.58 + 4.48i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (1.15 - 2.00i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 - 9.34iT - 41T^{2}$$
43 $$1 - 6.84iT - 43T^{2}$$
47 $$1 + (-3.12 - 1.80i)T + (23.5 + 40.7i)T^{2}$$
53 $$1 + (-0.0130 - 0.0225i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (-0.0160 + 0.00925i)T + (29.5 - 51.0i)T^{2}$$
61 $$1 + (-0.897 + 1.55i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (4.14 - 2.39i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 - 8.13iT - 71T^{2}$$
73 $$1 + (6.30 + 10.9i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (-13.5 - 7.81i)T + (39.5 + 68.4i)T^{2}$$
83 $$1 + 6.60T + 83T^{2}$$
89 $$1 + (1.31 + 0.760i)T + (44.5 + 77.0i)T^{2}$$
97 $$1 - 12.1T + 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

−11.46537875698653075714968843320, −10.61005117019823687295442961155, −9.785683918266457507153045152781, −8.770393635972966070086235120350, −7.64972677171583668201230378162, −6.64935720695290127403640013965, −6.24991515252308163816369438147, −4.61057036885500919340323643876, −2.15313003907106376160223005985, −1.04118334347645651100257501617, 1.84752280829492955116387295406, 3.39421570047353351594687728724, 5.31479766822787989034013089903, 5.92602049740697133987464512675, 7.25930393309060620568202422512, 8.737785097114788343020320103254, 9.212147884747128412938729705013, 10.27270994244341636254699009758, 10.85584317001534177858962409990, 11.66027208411296070418924990684