Properties

Label 2-11200-1.1-c1-0-30
Degree $2$
Conductor $11200$
Sign $1$
Analytic cond. $89.4324$
Root an. cond. $9.45687$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 3·9-s + 4·11-s + 2·13-s + 6·17-s − 8·19-s − 6·29-s + 8·31-s − 2·37-s + 2·41-s − 4·43-s + 8·47-s + 49-s + 6·53-s + 6·61-s − 3·63-s − 4·67-s − 8·71-s − 10·73-s + 4·77-s + 16·79-s + 9·81-s + 8·83-s − 6·89-s + 2·91-s + 6·97-s − 12·99-s + ⋯
L(s)  = 1  + 0.377·7-s − 9-s + 1.20·11-s + 0.554·13-s + 1.45·17-s − 1.83·19-s − 1.11·29-s + 1.43·31-s − 0.328·37-s + 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.768·61-s − 0.377·63-s − 0.488·67-s − 0.949·71-s − 1.17·73-s + 0.455·77-s + 1.80·79-s + 81-s + 0.878·83-s − 0.635·89-s + 0.209·91-s + 0.609·97-s − 1.20·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11200\)    =    \(2^{6} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(89.4324\)
Root analytic conductor: \(9.45687\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{11200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 11200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.212451677\)
\(L(\frac12)\) \(\approx\) \(2.212451677\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
good3 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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

−16.75041798472730, −15.98831177242494, −14.98466864517356, −14.89150201992704, −14.25907998992417, −13.73514115587297, −13.10415180472810, −12.26439827091722, −11.89286957424514, −11.35797103142126, −10.67957973324950, −10.17896892928625, −9.311973503077288, −8.742896895578126, −8.337962811166629, −7.639524921322695, −6.809930204595664, −6.097268854969746, −5.751533997350669, −4.825056897404686, −4.017798193147831, −3.486085970384816, −2.531301113920433, −1.659558674477443, −0.7075870714173326, 0.7075870714173326, 1.659558674477443, 2.531301113920433, 3.486085970384816, 4.017798193147831, 4.825056897404686, 5.751533997350669, 6.097268854969746, 6.809930204595664, 7.639524921322695, 8.337962811166629, 8.742896895578126, 9.311973503077288, 10.17896892928625, 10.67957973324950, 11.35797103142126, 11.89286957424514, 12.26439827091722, 13.10415180472810, 13.73514115587297, 14.25907998992417, 14.89150201992704, 14.98466864517356, 15.98831177242494, 16.75041798472730

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