Properties

Label 2-9200-1.1-c1-0-100
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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.43·3-s + 3.08·7-s − 0.950·9-s + 6.46·11-s − 3.95·13-s + 3.43·17-s − 3.08·19-s + 4.41·21-s − 23-s − 5.65·27-s + 0.863·29-s + 5.95·31-s + 9.26·33-s + 7.03·37-s − 5.65·39-s + 5.60·41-s + 8·43-s + 3.90·47-s + 2.53·49-s + 4.91·51-s + 6·53-s − 4.41·57-s − 6.86·59-s − 13.5·61-s − 2.93·63-s − 10.0·67-s − 1.43·69-s + ⋯
L(s)  = 1  + 0.826·3-s + 1.16·7-s − 0.316·9-s + 1.95·11-s − 1.09·13-s + 0.832·17-s − 0.708·19-s + 0.964·21-s − 0.208·23-s − 1.08·27-s + 0.160·29-s + 1.06·31-s + 1.61·33-s + 1.15·37-s − 0.905·39-s + 0.875·41-s + 1.21·43-s + 0.569·47-s + 0.361·49-s + 0.687·51-s + 0.824·53-s − 0.585·57-s − 0.893·59-s − 1.72·61-s − 0.369·63-s − 1.23·67-s − 0.172·69-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.645967222\)
\(L(\frac12)\) \(\approx\) \(3.645967222\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.43T + 3T^{2} \)
7 \( 1 - 3.08T + 7T^{2} \)
11 \( 1 - 6.46T + 11T^{2} \)
13 \( 1 + 3.95T + 13T^{2} \)
17 \( 1 - 3.43T + 17T^{2} \)
19 \( 1 + 3.08T + 19T^{2} \)
29 \( 1 - 0.863T + 29T^{2} \)
31 \( 1 - 5.95T + 31T^{2} \)
37 \( 1 - 7.03T + 37T^{2} \)
41 \( 1 - 5.60T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 3.90T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 6.86T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 + 2.56T + 71T^{2} \)
73 \( 1 + 5.90T + 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 - 9.03T + 83T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 - 14.2T + 97T^{2} \)
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

−7.61326386546542880552852489830, −7.42124561030534152760549846469, −6.21331645207634435205308619078, −5.85823561561743584192715713400, −4.56114083780800665708573259867, −4.40622848291496324132107726484, −3.40221396002625490752190775683, −2.57587795481510668665585275086, −1.82067198574363157588847723210, −0.927122802155948535013386344625, 0.927122802155948535013386344625, 1.82067198574363157588847723210, 2.57587795481510668665585275086, 3.40221396002625490752190775683, 4.40622848291496324132107726484, 4.56114083780800665708573259867, 5.85823561561743584192715713400, 6.21331645207634435205308619078, 7.42124561030534152760549846469, 7.61326386546542880552852489830

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