↑

Properties

Label 2-560-5.4-c1-0-1
Degree $2$
Conductor $560$
Sign $-0.994 - 0.100i$
Analytic cond. $4.47162$
Root an. cond. $2.11462$
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  + 2.44i·3-s + (−0.224 + 2.22i)5-s + i·7-s − 2.99·9-s − 4.89·11-s + 0.449i·13-s + (−5.44 − 0.550i)15-s + 2i·17-s + 6.44·19-s − 2.44·21-s − 6.89i·23-s + (−4.89 − i)25-s + 2.89·29-s + 0.898·31-s − 11.9i·33-s + ⋯
L(s)  = 1  + 1.41i·3-s + (−0.100 + 0.994i)5-s + 0.377i·7-s − 0.999·9-s − 1.47·11-s + 0.124i·13-s + (−1.40 − 0.142i)15-s + 0.485i·17-s + 1.47·19-s − 0.534·21-s − 1.43i·23-s + (−0.979 − 0.200i)25-s + 0.538·29-s + 0.161·31-s − 2.08i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.994 - 0.100i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.994 - 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0547101 + 1.08590i\)
\(L(\frac12)\) \(\approx\) \(0.0547101 + 1.08590i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.224 - 2.22i)T \)
7 \( 1 - iT \)
good3 \( 1 - 2.44iT - 3T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 - 0.449iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 6.44T + 19T^{2} \)
23 \( 1 + 6.89iT - 23T^{2} \)
29 \( 1 - 2.89T + 29T^{2} \)
31 \( 1 - 0.898T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 8.89iT - 43T^{2} \)
47 \( 1 - 0.898iT - 47T^{2} \)
53 \( 1 - 1.10iT - 53T^{2} \)
59 \( 1 + 6.44T + 59T^{2} \)
61 \( 1 - 8.44T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 - 6.89iT - 73T^{2} \)
79 \( 1 + 2.89T + 79T^{2} \)
83 \( 1 - 2.44iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 3.79iT - 97T^{2} \)
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

−10.94233245004098263325515437387, −10.19612968375096452621348887364, −9.846874011343112458534974745042, −8.616238143776493231826296875299, −7.76284298523440072796984466160, −6.57452826734288802278571971606, −5.44633409098404133479948268008, −4.65498515269625819021756396253, −3.40736505686345329035316448916, −2.62398767237310129532409966925, 0.61608851251157899507562863377, 1.86767772322833534227136792854, 3.30188167355794241111823405207, 4.99537457163729420102410260742, 5.61574650205583044526235799684, 6.99415327141256090395497325092, 7.68414523634273407074751041424, 8.213501166713106520238873691060, 9.379112251077911053384122580331, 10.30938474347751412150041228464

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