↑

Properties

Label 2-280-280.213-c1-0-8
Degree $2$
Conductor $280$
Sign $-0.556 - 0.830i$
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  + (1.28 + 0.592i)2-s + (−0.999 + 0.267i)3-s + (1.29 + 1.52i)4-s + (−2.16 + 0.546i)5-s + (−1.44 − 0.248i)6-s + (−2.32 + 1.26i)7-s + (0.763 + 2.72i)8-s + (−1.67 + 0.965i)9-s + (−3.10 − 0.582i)10-s + (4.88 + 2.81i)11-s + (−1.70 − 1.17i)12-s + (0.271 + 0.271i)13-s + (−3.73 + 0.240i)14-s + (2.01 − 1.12i)15-s + (−0.633 + 3.94i)16-s + (−1.95 + 0.524i)17-s + ⋯
L(s)  = 1  + (0.907 + 0.419i)2-s + (−0.576 + 0.154i)3-s + (0.648 + 0.761i)4-s + (−0.969 + 0.244i)5-s + (−0.588 − 0.101i)6-s + (−0.879 + 0.476i)7-s + (0.270 + 0.962i)8-s + (−0.557 + 0.321i)9-s + (−0.982 − 0.184i)10-s + (1.47 + 0.849i)11-s + (−0.491 − 0.338i)12-s + (0.0753 + 0.0753i)13-s + (−0.997 + 0.0643i)14-s + (0.521 − 0.290i)15-s + (−0.158 + 0.987i)16-s + (−0.474 + 0.127i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.556 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.556 - 0.830i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.556 - 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.620181 + 1.16245i\)
\(L(\frac12)\) \(\approx\) \(0.620181 + 1.16245i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 - 0.592i)T \)
5 \( 1 + (2.16 - 0.546i)T \)
7 \( 1 + (2.32 - 1.26i)T \)
good3 \( 1 + (0.999 - 0.267i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-4.88 - 2.81i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.271 - 0.271i)T + 13iT^{2} \)
17 \( 1 + (1.95 - 0.524i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.47 + 1.42i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.58 + 5.90i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 4.06T + 29T^{2} \)
31 \( 1 + (-5.81 - 3.35i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.133 - 0.499i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.88iT - 41T^{2} \)
43 \( 1 + (2.46 - 2.46i)T - 43iT^{2} \)
47 \( 1 + (1.00 - 3.75i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.18 - 4.43i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-9.09 - 5.24i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.80 + 8.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.0 + 3.21i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.12T + 71T^{2} \)
73 \( 1 + (0.996 + 3.72i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.39 + 0.804i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.22 + 3.22i)T + 83iT^{2} \)
89 \( 1 + (-8.78 - 15.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.95 + 1.95i)T + 97iT^{2} \)
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

−12.06755184776589336091064154290, −11.63415935150171816651024165676, −10.68335399788327604288612472782, −9.213938474785492013373747785448, −8.158980432429689410003619705927, −6.80697297861989451954608577241, −6.40366372450357276815456737878, −4.98535654408435278969453004219, −4.02819334483272130653098470852, −2.79450864345042953235018641486, 0.832632136162050772402643542425, 3.34434992064165002090838700571, 3.93757925820242496347845921084, 5.44831458623049192676433168714, 6.40923563396775217835323017350, 7.20185593000779791996373539329, 8.795892692232454374021427672341, 9.821689912131937513171941240342, 11.22063601494012067553695762684, 11.52984903777518450512097941736

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