Properties

Label 2-9200-1.1-c1-0-153
Degree $2$
Conductor $9200$
Sign $-1$
Analytic cond. $73.4623$
Root an. cond. $8.57101$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.873·3-s − 0.992·7-s − 2.23·9-s − 1.83·11-s − 3.28·13-s + 6.63·17-s + 5.64·19-s − 0.866·21-s − 23-s − 4.57·27-s + 2.01·29-s + 0.315·31-s − 1.60·33-s + 3.07·37-s − 2.87·39-s − 1.34·41-s − 5.97·43-s + 0.306·47-s − 6.01·49-s + 5.79·51-s + 6.98·53-s + 4.92·57-s − 9.49·59-s + 5.56·61-s + 2.22·63-s + 0.853·67-s − 0.873·69-s + ⋯
L(s)  = 1  + 0.504·3-s − 0.375·7-s − 0.745·9-s − 0.552·11-s − 0.911·13-s + 1.60·17-s + 1.29·19-s − 0.189·21-s − 0.208·23-s − 0.880·27-s + 0.374·29-s + 0.0565·31-s − 0.278·33-s + 0.505·37-s − 0.459·39-s − 0.210·41-s − 0.911·43-s + 0.0446·47-s − 0.859·49-s + 0.811·51-s + 0.959·53-s + 0.652·57-s − 1.23·59-s + 0.712·61-s + 0.279·63-s + 0.104·67-s − 0.105·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9200\)    =    \(2^{4} \cdot 5^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(73.4623\)
Root analytic conductor: \(8.57101\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{9200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9200,\ (\ :1/2),\ -1)\)

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 \( 1 \)
23 \( 1 + T \)
good3 \( 1 - 0.873T + 3T^{2} \)
7 \( 1 + 0.992T + 7T^{2} \)
11 \( 1 + 1.83T + 11T^{2} \)
13 \( 1 + 3.28T + 13T^{2} \)
17 \( 1 - 6.63T + 17T^{2} \)
19 \( 1 - 5.64T + 19T^{2} \)
29 \( 1 - 2.01T + 29T^{2} \)
31 \( 1 - 0.315T + 31T^{2} \)
37 \( 1 - 3.07T + 37T^{2} \)
41 \( 1 + 1.34T + 41T^{2} \)
43 \( 1 + 5.97T + 43T^{2} \)
47 \( 1 - 0.306T + 47T^{2} \)
53 \( 1 - 6.98T + 53T^{2} \)
59 \( 1 + 9.49T + 59T^{2} \)
61 \( 1 - 5.56T + 61T^{2} \)
67 \( 1 - 0.853T + 67T^{2} \)
71 \( 1 + 0.797T + 71T^{2} \)
73 \( 1 + 7.67T + 73T^{2} \)
79 \( 1 - 3.62T + 79T^{2} \)
83 \( 1 + 17.1T + 83T^{2} \)
89 \( 1 + 7.01T + 89T^{2} \)
97 \( 1 - 18.5T + 97T^{2} \)
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

−7.59817892856172960228505947823, −6.81593052408185734305323217419, −5.83058803458323559132033830413, −5.41162214932000678099396729915, −4.68224131238102727389529148433, −3.53395707412197133009535011486, −3.06522632654250207599768448666, −2.42823033162138933556932225916, −1.23821922239794968829086237943, 0, 1.23821922239794968829086237943, 2.42823033162138933556932225916, 3.06522632654250207599768448666, 3.53395707412197133009535011486, 4.68224131238102727389529148433, 5.41162214932000678099396729915, 5.83058803458323559132033830413, 6.81593052408185734305323217419, 7.59817892856172960228505947823

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