↑

Properties

Label 2-280-8.5-c1-0-20
Degree $2$
Conductor $280$
Sign $-0.949 - 0.312i$
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  + (−1.14 − 0.832i)2-s − 3.42i·3-s + (0.613 + 1.90i)4-s − i·5-s + (−2.85 + 3.91i)6-s + 7-s + (0.883 − 2.68i)8-s − 8.75·9-s + (−0.832 + 1.14i)10-s − 4.42i·11-s + (6.52 − 2.10i)12-s + 0.766i·13-s + (−1.14 − 0.832i)14-s − 3.42·15-s + (−3.24 + 2.33i)16-s − 0.356·17-s + ⋯
L(s)  = 1  + (−0.808 − 0.588i)2-s − 1.97i·3-s + (0.306 + 0.951i)4-s − 0.447i·5-s + (−1.16 + 1.59i)6-s + 0.377·7-s + (0.312 − 0.949i)8-s − 2.91·9-s + (−0.263 + 0.361i)10-s − 1.33i·11-s + (1.88 − 0.607i)12-s + 0.212i·13-s + (−0.305 − 0.222i)14-s − 0.885·15-s + (−0.811 + 0.583i)16-s − 0.0864·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.949 - 0.312i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.949 - 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.119373 + 0.745219i\)
\(L(\frac12)\) \(\approx\) \(0.119373 + 0.745219i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.14 + 0.832i)T \)
5 \( 1 + iT \)
7 \( 1 - T \)
good3 \( 1 + 3.42iT - 3T^{2} \)
11 \( 1 + 4.42iT - 11T^{2} \)
13 \( 1 - 0.766iT - 13T^{2} \)
17 \( 1 + 0.356T + 17T^{2} \)
19 \( 1 - 3.20iT - 19T^{2} \)
23 \( 1 - 4.70T + 23T^{2} \)
29 \( 1 - 4.05iT - 29T^{2} \)
31 \( 1 - 2.12T + 31T^{2} \)
37 \( 1 + 8.70iT - 37T^{2} \)
41 \( 1 + 3.95T + 41T^{2} \)
43 \( 1 + 5.28iT - 43T^{2} \)
47 \( 1 - 3.32T + 47T^{2} \)
53 \( 1 + 12.3iT - 53T^{2} \)
59 \( 1 + 5.63iT - 59T^{2} \)
61 \( 1 - 10.9iT - 61T^{2} \)
67 \( 1 + 0.0448iT - 67T^{2} \)
71 \( 1 + 4.27T + 71T^{2} \)
73 \( 1 + 2.27T + 73T^{2} \)
79 \( 1 - 6.43T + 79T^{2} \)
83 \( 1 + 9.61iT - 83T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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

−11.51319823836264106216585131728, −10.73718330576611313039166857238, −8.947750057457729105795053008460, −8.481060599426369924434834825408, −7.62150598115597365464762645201, −6.72223977420141505773878304150, −5.58656246850129030454289023363, −3.28869530291047926759917716443, −1.93348288530510566445666495196, −0.75383562473192808647176412553, 2.73331384653930307915963180402, 4.46234523458957439582682523670, 5.13876710776114982022295431304, 6.42910556924296532337111093263, 7.76113478302951754776980362195, 8.848707703127719351186544606772, 9.608756396650852374072221608016, 10.26704475289789136307814870458, 10.98677506063343068141916753454, 11.78895374434477195097230604282

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