Properties

Label 2-9200-1.1-c1-0-138
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  − 0.580·3-s + 0.315·7-s − 2.66·9-s + 4.34·11-s − 4.58·13-s − 0.917·17-s − 2.76·19-s − 0.182·21-s − 23-s + 3.28·27-s + 7.03·29-s + 0.867·31-s − 2.52·33-s − 4.68·37-s + 2.65·39-s + 4.69·41-s + 9.08·43-s + 8.24·47-s − 6.90·49-s + 0.532·51-s − 10.9·53-s + 1.60·57-s − 1.50·59-s + 11.4·61-s − 0.839·63-s − 1.68·67-s + 0.580·69-s + ⋯
L(s)  = 1  − 0.335·3-s + 0.119·7-s − 0.887·9-s + 1.31·11-s − 1.27·13-s − 0.222·17-s − 0.634·19-s − 0.0399·21-s − 0.208·23-s + 0.632·27-s + 1.30·29-s + 0.155·31-s − 0.439·33-s − 0.770·37-s + 0.425·39-s + 0.732·41-s + 1.38·43-s + 1.20·47-s − 0.985·49-s + 0.0745·51-s − 1.49·53-s + 0.212·57-s − 0.195·59-s + 1.46·61-s − 0.105·63-s − 0.205·67-s + 0.0698·69-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: \(1\)
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 \( 1 + 0.580T + 3T^{2} \)
7 \( 1 - 0.315T + 7T^{2} \)
11 \( 1 - 4.34T + 11T^{2} \)
13 \( 1 + 4.58T + 13T^{2} \)
17 \( 1 + 0.917T + 17T^{2} \)
19 \( 1 + 2.76T + 19T^{2} \)
29 \( 1 - 7.03T + 29T^{2} \)
31 \( 1 - 0.867T + 31T^{2} \)
37 \( 1 + 4.68T + 37T^{2} \)
41 \( 1 - 4.69T + 41T^{2} \)
43 \( 1 - 9.08T + 43T^{2} \)
47 \( 1 - 8.24T + 47T^{2} \)
53 \( 1 + 10.9T + 53T^{2} \)
59 \( 1 + 1.50T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 1.68T + 67T^{2} \)
71 \( 1 - 5.36T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 + 7.39T + 79T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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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

−7.22451219551922048091479891456, −6.67176897784629607543062159514, −6.06347138091190296114021554614, −5.34092382542241090721270437874, −4.56699531362554573517798088234, −3.99797599399550975876531048676, −2.92335430230300634754465166924, −2.27950654838217029671438233506, −1.13870652760093531517582876509, 0, 1.13870652760093531517582876509, 2.27950654838217029671438233506, 2.92335430230300634754465166924, 3.99797599399550975876531048676, 4.56699531362554573517798088234, 5.34092382542241090721270437874, 6.06347138091190296114021554614, 6.67176897784629607543062159514, 7.22451219551922048091479891456

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