Properties

Label 2-280-280.67-c1-0-35
Degree $2$
Conductor $280$
Sign $-0.519 + 0.854i$
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  + (−1.16 + 0.804i)2-s + (−0.540 − 2.01i)3-s + (0.706 − 1.87i)4-s + (0.161 − 2.23i)5-s + (2.25 + 1.91i)6-s + (1.67 − 2.04i)7-s + (0.683 + 2.74i)8-s + (−1.17 + 0.679i)9-s + (1.60 + 2.72i)10-s + (−2.35 + 4.08i)11-s + (−4.15 − 0.413i)12-s + (−3.68 − 3.68i)13-s + (−0.298 + 3.72i)14-s + (−4.58 + 0.879i)15-s + (−3.00 − 2.64i)16-s + (5.98 − 1.60i)17-s + ⋯
L(s)  = 1  + (−0.822 + 0.568i)2-s + (−0.311 − 1.16i)3-s + (0.353 − 0.935i)4-s + (0.0721 − 0.997i)5-s + (0.918 + 0.780i)6-s + (0.632 − 0.774i)7-s + (0.241 + 0.970i)8-s + (−0.392 + 0.226i)9-s + (0.507 + 0.861i)10-s + (−0.711 + 1.23i)11-s + (−1.19 − 0.119i)12-s + (−1.02 − 1.02i)13-s + (−0.0796 + 0.996i)14-s + (−1.18 + 0.227i)15-s + (−0.750 − 0.660i)16-s + (1.45 − 0.389i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.519 + 0.854i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.519 + 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.350169 - 0.622736i\)
\(L(\frac12)\) \(\approx\) \(0.350169 - 0.622736i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 - 0.804i)T \)
5 \( 1 + (-0.161 + 2.23i)T \)
7 \( 1 + (-1.67 + 2.04i)T \)
good3 \( 1 + (0.540 + 2.01i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (2.35 - 4.08i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.68 + 3.68i)T + 13iT^{2} \)
17 \( 1 + (-5.98 + 1.60i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.62 - 1.51i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.371 - 1.38i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.08T + 29T^{2} \)
31 \( 1 + (-0.111 - 0.0644i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.12 - 0.568i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 1.37T + 41T^{2} \)
43 \( 1 + (-2.05 + 2.05i)T - 43iT^{2} \)
47 \( 1 + (11.0 + 2.96i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.82 - 0.758i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-11.1 - 6.42i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.49 + 3.17i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.28 + 1.95i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.96iT - 71T^{2} \)
73 \( 1 + (-0.205 - 0.768i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.00532 + 0.00922i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.27 + 4.27i)T - 83iT^{2} \)
89 \( 1 + (-6.15 + 3.55i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.22 - 8.22i)T + 97iT^{2} \)
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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

−11.69890676839991951677803465234, −10.22483593368511785851177596557, −9.793680141225219427086818050636, −8.125563150701034832152257301583, −7.75016244554014522594120100572, −7.00003144797781665987958555648, −5.57948909683941193171617424821, −4.79155149315795994701721893913, −1.97279232213947278667461752493, −0.73995300224962442771154111254, 2.37281138467943191414013113299, 3.52886981363795947237339099125, 4.91521936875975974095538790244, 6.21814119013282252674914921853, 7.61862597488513330482060306993, 8.545799060836225515015131450227, 9.676041376288810321904334617728, 10.25088510867879273784801913454, 11.16182230154842535651975180164, 11.57258333834632449326666202455

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