Properties

Label 2-9200-1.1-c1-0-135
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  − 3-s − 2·9-s − 2·11-s + 5·13-s + 4·17-s + 2·19-s − 23-s + 5·27-s − 3·29-s − 7·31-s + 2·33-s + 2·37-s − 5·39-s − 9·41-s − 4·43-s − 9·47-s − 7·49-s − 4·51-s + 6·53-s − 2·57-s + 2·61-s − 2·67-s + 69-s + 71-s − 73-s + 14·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s − 0.603·11-s + 1.38·13-s + 0.970·17-s + 0.458·19-s − 0.208·23-s + 0.962·27-s − 0.557·29-s − 1.25·31-s + 0.348·33-s + 0.328·37-s − 0.800·39-s − 1.40·41-s − 0.609·43-s − 1.31·47-s − 49-s − 0.560·51-s + 0.824·53-s − 0.264·57-s + 0.256·61-s − 0.244·67-s + 0.120·69-s + 0.118·71-s − 0.117·73-s + 1.57·79-s + 1/9·81-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 \( 1 + T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−7.42314403120293093723401726142, −6.50857844199842191639699083912, −5.99121360965929980573080081366, −5.33701554244838769728087900507, −4.87338196650952481178472651512, −3.51743385846087025954752613129, −3.37382764401319970889447237756, −2.09785343188684447011933375493, −1.13084125980465834121328993606, 0, 1.13084125980465834121328993606, 2.09785343188684447011933375493, 3.37382764401319970889447237756, 3.51743385846087025954752613129, 4.87338196650952481178472651512, 5.33701554244838769728087900507, 5.99121360965929980573080081366, 6.50857844199842191639699083912, 7.42314403120293093723401726142

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