Properties

Label 2-9200-1.1-c1-0-198
Degree $2$
Conductor $9200$
Sign $-1$
Analytic cond. $73.4623$
Root an. cond. $8.57101$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 3·7-s + 9-s − 6·13-s − 7·17-s + 4·19-s + 6·21-s + 23-s − 4·27-s − 9·29-s + 3·31-s − 7·37-s − 12·39-s + 9·41-s + 4·43-s + 2·47-s + 2·49-s − 14·51-s − 7·53-s + 8·57-s − 9·59-s − 2·61-s + 3·63-s − 13·67-s + 2·69-s + 13·71-s − 4·73-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.13·7-s + 1/3·9-s − 1.66·13-s − 1.69·17-s + 0.917·19-s + 1.30·21-s + 0.208·23-s − 0.769·27-s − 1.67·29-s + 0.538·31-s − 1.15·37-s − 1.92·39-s + 1.40·41-s + 0.609·43-s + 0.291·47-s + 2/7·49-s − 1.96·51-s − 0.961·53-s + 1.05·57-s − 1.17·59-s − 0.256·61-s + 0.377·63-s − 1.58·67-s + 0.240·69-s + 1.54·71-s − 0.468·73-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: yes
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 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 7 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−7.52246097639366876208254473435, −7.05498017301087873222996716119, −5.96415214501196404120354284803, −5.09117471152249895096623105734, −4.60512350573298215100320611453, −3.82689952300772687711030084359, −2.83183807701483073477843055711, −2.28457606314058353081220807317, −1.59013445025816081356170839496, 0, 1.59013445025816081356170839496, 2.28457606314058353081220807317, 2.83183807701483073477843055711, 3.82689952300772687711030084359, 4.60512350573298215100320611453, 5.09117471152249895096623105734, 5.96415214501196404120354284803, 7.05498017301087873222996716119, 7.52246097639366876208254473435

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