Properties

Label 2-4400-1.1-c1-0-84
Degree $2$
Conductor $4400$
Sign $-1$
Analytic cond. $35.1341$
Root an. cond. $5.92740$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·7-s − 2·9-s + 11-s + 4·13-s − 6·17-s − 8·19-s + 2·21-s − 3·23-s − 5·27-s − 5·31-s + 33-s + 37-s + 4·39-s − 10·43-s − 3·49-s − 6·51-s + 6·53-s − 8·57-s − 3·59-s − 4·61-s − 4·63-s − 67-s − 3·69-s − 15·71-s + 4·73-s + 2·77-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s − 2/3·9-s + 0.301·11-s + 1.10·13-s − 1.45·17-s − 1.83·19-s + 0.436·21-s − 0.625·23-s − 0.962·27-s − 0.898·31-s + 0.174·33-s + 0.164·37-s + 0.640·39-s − 1.52·43-s − 3/7·49-s − 0.840·51-s + 0.824·53-s − 1.05·57-s − 0.390·59-s − 0.512·61-s − 0.503·63-s − 0.122·67-s − 0.361·69-s − 1.78·71-s + 0.468·73-s + 0.227·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4400\)    =    \(2^{4} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(35.1341\)
Root analytic conductor: \(5.92740\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4400,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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

−8.270291326594430363325745498073, −7.38829777934131723608792260408, −6.39194712026754517361415727782, −6.00643085529535229889961932546, −4.86824029564301859859497168572, −4.15869750405124088599031875635, −3.43029130494338377492333953612, −2.27940290108639207983747311479, −1.70082535066997228503423935134, 0, 1.70082535066997228503423935134, 2.27940290108639207983747311479, 3.43029130494338377492333953612, 4.15869750405124088599031875635, 4.86824029564301859859497168572, 6.00643085529535229889961932546, 6.39194712026754517361415727782, 7.38829777934131723608792260408, 8.270291326594430363325745498073

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