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Properties

Label 2-280-56.3-c1-0-7
Degree $2$
Conductor $280$
Sign $-0.451 - 0.892i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.394 + 1.35i)2-s + (1.75 + 1.01i)3-s + (−1.68 − 1.07i)4-s + (0.5 + 0.866i)5-s + (−2.06 + 1.98i)6-s + (1.63 + 2.07i)7-s + (2.12 − 1.86i)8-s + (0.552 + 0.957i)9-s + (−1.37 + 0.336i)10-s + (−0.572 + 0.991i)11-s + (−1.87 − 3.59i)12-s + 0.714·13-s + (−3.46 + 1.40i)14-s + 2.02i·15-s + (1.69 + 3.62i)16-s + (−1.98 − 1.14i)17-s + ⋯
L(s)  = 1  + (−0.279 + 0.960i)2-s + (1.01 + 0.584i)3-s + (−0.843 − 0.536i)4-s + (0.223 + 0.387i)5-s + (−0.844 + 0.809i)6-s + (0.619 + 0.785i)7-s + (0.750 − 0.660i)8-s + (0.184 + 0.319i)9-s + (−0.434 + 0.106i)10-s + (−0.172 + 0.299i)11-s + (−0.541 − 1.03i)12-s + 0.198·13-s + (−0.926 + 0.375i)14-s + 0.523i·15-s + (0.424 + 0.905i)16-s + (−0.480 − 0.277i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.451 - 0.892i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.451 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.777496 + 1.26552i\)
\(L(\frac12)\) \(\approx\) \(0.777496 + 1.26552i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.394 - 1.35i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-1.63 - 2.07i)T \)
good3 \( 1 + (-1.75 - 1.01i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.572 - 0.991i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.714T + 13T^{2} \)
17 \( 1 + (1.98 + 1.14i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.36 - 1.94i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.00 + 2.31i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.54iT - 29T^{2} \)
31 \( 1 + (-0.590 + 1.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.72 - 3.30i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.9iT - 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + (2.60 + 4.51i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.25 + 1.30i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.5 - 6.07i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.13 - 12.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.08 + 7.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.8iT - 71T^{2} \)
73 \( 1 + (4.71 + 2.72i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.20 + 4.73i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.78iT - 83T^{2} \)
89 \( 1 + (11.0 - 6.38i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.18iT - 97T^{2} \)
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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

−12.30192980837439796317242829222, −10.86689896547200209682787158538, −9.984883857135290608342867342510, −8.930212685239043528788970728058, −8.586209154841957978327026135846, −7.46038110400273581607806431226, −6.28534028133670193891502515239, −5.12518538539941992552028359437, −3.95986726759840117900774511533, −2.34107107464017420347527152053, 1.33315747738696495727544556301, 2.56506664643122097158223956345, 3.89556345847262608271673524353, 5.09003366452998820867796988213, 7.01410173775255249004956570377, 8.092236573816307129207484786523, 8.613849064749751482524546392271, 9.562087084741015389891947054643, 10.78312980034471179546264540299, 11.33101790853348325516502425910

Graph of the $Z$-function along the critical line