Properties

Label 2-2200-1.1-c1-0-16
Degree $2$
Conductor $2200$
Sign $1$
Analytic cond. $17.5670$
Root an. cond. $4.19131$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·7-s + 9-s − 11-s + 4·17-s + 4·19-s − 4·21-s + 6·23-s − 4·27-s + 2·29-s + 8·31-s − 2·33-s + 4·37-s − 6·41-s + 6·43-s + 2·47-s − 3·49-s + 8·51-s + 12·53-s + 8·57-s + 4·59-s + 14·61-s − 2·63-s − 10·67-s + 12·69-s + 8·71-s − 4·73-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.970·17-s + 0.917·19-s − 0.872·21-s + 1.25·23-s − 0.769·27-s + 0.371·29-s + 1.43·31-s − 0.348·33-s + 0.657·37-s − 0.937·41-s + 0.914·43-s + 0.291·47-s − 3/7·49-s + 1.12·51-s + 1.64·53-s + 1.05·57-s + 0.520·59-s + 1.79·61-s − 0.251·63-s − 1.22·67-s + 1.44·69-s + 0.949·71-s − 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(17.5670\)
Root analytic conductor: \(4.19131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.528623612\)
\(L(\frac12)\) \(\approx\) \(2.528623612\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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

−8.958936033637511219237653499509, −8.384419832698018317161663115322, −7.58735384572792137844530015224, −6.93573815786884005364904081442, −5.90459996973399716388407138086, −5.07107578009454333641284586951, −3.88810016729030851806993977453, −3.07092255284723312597045006148, −2.57028271253547471540283448852, −1.01744955324563883925903459776, 1.01744955324563883925903459776, 2.57028271253547471540283448852, 3.07092255284723312597045006148, 3.88810016729030851806993977453, 5.07107578009454333641284586951, 5.90459996973399716388407138086, 6.93573815786884005364904081442, 7.58735384572792137844530015224, 8.384419832698018317161663115322, 8.958936033637511219237653499509

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