Properties

Label 2-9200-1.1-c1-0-176
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.40·3-s − 4.41·7-s + 2.76·9-s − 2.29·11-s + 6.92·13-s + 1.51·17-s − 2.89·19-s − 10.5·21-s − 23-s − 0.570·27-s − 7.68·29-s − 3.85·31-s − 5.50·33-s + 8.62·37-s + 16.6·39-s − 6.44·41-s + 3.48·43-s + 6.19·47-s + 12.4·49-s + 3.63·51-s + 2.17·53-s − 6.95·57-s + 11.7·59-s − 5.11·61-s − 12.1·63-s − 9.94·67-s − 2.40·69-s + ⋯
L(s)  = 1  + 1.38·3-s − 1.66·7-s + 0.920·9-s − 0.691·11-s + 1.92·13-s + 0.367·17-s − 0.665·19-s − 2.31·21-s − 0.208·23-s − 0.109·27-s − 1.42·29-s − 0.692·31-s − 0.958·33-s + 1.41·37-s + 2.66·39-s − 1.00·41-s + 0.531·43-s + 0.903·47-s + 1.78·49-s + 0.508·51-s + 0.299·53-s − 0.921·57-s + 1.53·59-s − 0.654·61-s − 1.53·63-s − 1.21·67-s − 0.288·69-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 \( 1 - 2.40T + 3T^{2} \)
7 \( 1 + 4.41T + 7T^{2} \)
11 \( 1 + 2.29T + 11T^{2} \)
13 \( 1 - 6.92T + 13T^{2} \)
17 \( 1 - 1.51T + 17T^{2} \)
19 \( 1 + 2.89T + 19T^{2} \)
29 \( 1 + 7.68T + 29T^{2} \)
31 \( 1 + 3.85T + 31T^{2} \)
37 \( 1 - 8.62T + 37T^{2} \)
41 \( 1 + 6.44T + 41T^{2} \)
43 \( 1 - 3.48T + 43T^{2} \)
47 \( 1 - 6.19T + 47T^{2} \)
53 \( 1 - 2.17T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 + 5.11T + 61T^{2} \)
67 \( 1 + 9.94T + 67T^{2} \)
71 \( 1 + 3.41T + 71T^{2} \)
73 \( 1 + 8.95T + 73T^{2} \)
79 \( 1 - 1.92T + 79T^{2} \)
83 \( 1 + 8.04T + 83T^{2} \)
89 \( 1 + 1.09T + 89T^{2} \)
97 \( 1 + 16.9T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.50181606353906745017633154404, −6.77904577168746900117172698042, −5.98119948619663594923359414121, −5.60304267731364295174122109762, −4.01654281754674569415451836006, −3.81393877548382263420594682735, −3.02948467624602344359305953154, −2.48124212432530265462058395912, −1.40815048300410945079438049643, 0, 1.40815048300410945079438049643, 2.48124212432530265462058395912, 3.02948467624602344359305953154, 3.81393877548382263420594682735, 4.01654281754674569415451836006, 5.60304267731364295174122109762, 5.98119948619663594923359414121, 6.77904577168746900117172698042, 7.50181606353906745017633154404

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