Properties

Label 2-280-7.2-c1-0-7
Degree $2$
Conductor $280$
Sign $-0.984 + 0.173i$
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  + (−1.29 − 2.23i)3-s + (−0.5 + 0.866i)5-s + (−0.292 − 2.62i)7-s + (−1.83 + 3.18i)9-s + (−0.839 − 1.45i)11-s − 4.84·13-s + 2.58·15-s + (−1 − 1.73i)17-s + (−3.42 + 5.92i)19-s + (−5.50 + 4.05i)21-s + (1.13 − 1.95i)23-s + (−0.499 − 0.866i)25-s + 1.75·27-s + 3.32·29-s + (−4.58 − 7.94i)31-s + ⋯
L(s)  = 1  + (−0.746 − 1.29i)3-s + (−0.223 + 0.387i)5-s + (−0.110 − 0.993i)7-s + (−0.613 + 1.06i)9-s + (−0.253 − 0.438i)11-s − 1.34·13-s + 0.667·15-s + (−0.242 − 0.420i)17-s + (−0.785 + 1.36i)19-s + (−1.20 + 0.884i)21-s + (0.235 − 0.408i)23-s + (−0.0999 − 0.173i)25-s + 0.337·27-s + 0.616·29-s + (−0.823 − 1.42i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.984 + 0.173i$
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: \((2,\ 280,\ (\ :1/2),\ -0.984 + 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0480615 - 0.551008i\)
\(L(\frac12)\) \(\approx\) \(0.0480615 - 0.551008i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.292 + 2.62i)T \)
good3 \( 1 + (1.29 + 2.23i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.839 + 1.45i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.84T + 13T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.42 - 5.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.13 + 1.95i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.32T + 29T^{2} \)
31 \( 1 + (4.58 + 7.94i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.42 + 2.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.52T + 41T^{2} \)
43 \( 1 - 6.58T + 43T^{2} \)
47 \( 1 + (-6.10 + 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.74 + 6.48i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.24 + 5.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.87 - 4.98i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-5.84 - 10.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.84 - 4.93i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 + (-2.92 + 5.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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

−11.45600439630618942355857030118, −10.72725020271821621800707018053, −9.721985932483798865040004331826, −8.047109763001687147315687759710, −7.37600063901439103038055379399, −6.62543321457133270199307982323, −5.59150258073620404340717824829, −4.09076195870915743930351395785, −2.29818703879983319749571464018, −0.43678025571468972624121645034, 2.68089200758246309366403551718, 4.43074120077593723920974351735, 4.98945116442219325126059479041, 6.02983673712212829535278577653, 7.42612207343484028981205389937, 8.927216585895992864730906058665, 9.414425640575757529453898702209, 10.50048207077455719484642455523, 11.22215599826273819961369679950, 12.26935787138324457670064917027

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