Properties

Label 2-280-7.2-c1-0-4
Degree $2$
Conductor $280$
Sign $0.827 + 0.561i$
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  + (−0.207 − 0.358i)3-s + (−0.5 + 0.866i)5-s + (1.62 − 2.09i)7-s + (1.41 − 2.44i)9-s + (−0.414 − 0.717i)11-s + 2·13-s + 0.414·15-s + (3.82 + 6.63i)17-s + (2.82 − 4.89i)19-s + (−1.08 − 0.148i)21-s + (2.79 − 4.83i)23-s + (−0.499 − 0.866i)25-s − 2.41·27-s − 7.82·29-s + (−0.414 − 0.717i)31-s + ⋯
L(s)  = 1  + (−0.119 − 0.207i)3-s + (−0.223 + 0.387i)5-s + (0.612 − 0.790i)7-s + (0.471 − 0.816i)9-s + (−0.124 − 0.216i)11-s + 0.554·13-s + 0.106·15-s + (0.928 + 1.60i)17-s + (0.648 − 1.12i)19-s + (−0.236 − 0.0324i)21-s + (0.582 − 1.00i)23-s + (−0.0999 − 0.173i)25-s − 0.464·27-s − 1.45·29-s + (−0.0743 − 0.128i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25352 - 0.385170i\)
\(L(\frac12)\) \(\approx\) \(1.25352 - 0.385170i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-1.62 + 2.09i)T \)
good3 \( 1 + (0.207 + 0.358i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.414 + 0.717i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-3.82 - 6.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.82 + 4.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.79 + 4.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.82T + 29T^{2} \)
31 \( 1 + (0.414 + 0.717i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.82 - 4.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.82T + 41T^{2} \)
43 \( 1 + 6.89T + 43T^{2} \)
47 \( 1 + (5.82 - 10.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.82 - 4.89i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.32 - 5.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.44 - 11.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (1.82 + 3.16i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.75T + 83T^{2} \)
89 \( 1 + (-2.67 + 4.62i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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

−11.64029908082314514060507090430, −10.88309042528597996546691812297, −10.06917782643637430410491447252, −8.829637360490311665116719758458, −7.77056097420570372136053467131, −6.92504814515629327873159953462, −5.88126113513778346760081005637, −4.38171337426815920232364466510, −3.34266005373798626897546140278, −1.24851763638738315928648747943, 1.75438787182776435296767405066, 3.52318138277711229973587567654, 5.06629752891740358246001047760, 5.50931223384563075237847348737, 7.33317776428191700037641836602, 7.990939318172307776076728691178, 9.175301682137778499968945261450, 9.970958941302115474549782591068, 11.22509027883141030294383322688, 11.78728730566358391678772207441

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