↑

Properties

Label 2-280-56.3-c1-0-6
Degree $2$
Conductor $280$
Sign $0.809 - 0.587i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.967 − 1.03i)2-s + (2.26 + 1.30i)3-s + (−0.129 + 1.99i)4-s + (0.5 + 0.866i)5-s + (−0.841 − 3.60i)6-s + (−1.72 + 2.00i)7-s + (2.18 − 1.79i)8-s + (1.92 + 3.33i)9-s + (0.410 − 1.35i)10-s + (0.530 − 0.919i)11-s + (−2.90 + 4.35i)12-s − 0.831·13-s + (3.73 − 0.157i)14-s + 2.61i·15-s + (−3.96 − 0.516i)16-s + (4.14 + 2.39i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.729i)2-s + (1.30 + 0.755i)3-s + (−0.0647 + 0.997i)4-s + (0.223 + 0.387i)5-s + (−0.343 − 1.47i)6-s + (−0.652 + 0.757i)7-s + (0.772 − 0.635i)8-s + (0.642 + 1.11i)9-s + (0.129 − 0.428i)10-s + (0.160 − 0.277i)11-s + (−0.838 + 1.25i)12-s − 0.230·13-s + (0.999 − 0.0422i)14-s + 0.675i·15-s + (−0.991 − 0.129i)16-s + (1.00 + 0.580i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.809 - 0.587i$
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: \((2,\ 280,\ (\ :1/2),\ 0.809 - 0.587i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26288 + 0.409978i\)
\(L(\frac12)\) \(\approx\) \(1.26288 + 0.409978i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.967 + 1.03i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (1.72 - 2.00i)T \)
good3 \( 1 + (-2.26 - 1.30i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-0.530 + 0.919i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.831T + 13T^{2} \)
17 \( 1 + (-4.14 - 2.39i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.03 - 1.17i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.32 - 0.763i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.07iT - 29T^{2} \)
31 \( 1 + (-4.78 + 8.28i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-10.0 + 5.81i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.76iT - 41T^{2} \)
43 \( 1 + 4.99T + 43T^{2} \)
47 \( 1 + (-1.75 - 3.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.61 - 3.81i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.31 + 1.91i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.51 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.36 - 12.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.99iT - 71T^{2} \)
73 \( 1 + (7.06 + 4.07i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.17 - 4.72i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.7iT - 83T^{2} \)
89 \( 1 + (-10.1 + 5.88i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.01iT - 97T^{2} \)
show more
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

−11.88280562008527181264525541710, −10.64570650451308790090170692127, −9.854642566004315369883847132488, −9.291001427446181258465024797864, −8.444443674759729137263986468808, −7.58834548739543401737028895468, −6.01615238897742374900108895344, −4.10391084687932068623286004813, −3.17509073587659525259508618763, −2.26312272512718789256785409344, 1.24422025368252699222136813123, 2.86615445946790354041755875589, 4.59014924071109187681474435097, 6.23269010028124475181572826286, 7.14263926501401356948304634622, 7.889353436279887045387636566714, 8.726401688692898860616929399425, 9.658776500210857194373262693452, 10.25516525900621702143519217018, 11.89885224145702247013256364902

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