Properties

Label 2-9200-1.1-c1-0-192
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 + 7-s + 9-s − 2·13-s + 5·17-s − 8·19-s + 2·21-s − 23-s − 4·27-s − 5·29-s + 5·31-s − 7·37-s − 4·39-s − 7·41-s + 4·43-s − 2·47-s − 6·49-s + 10·51-s + 53-s − 16·57-s − 3·59-s − 6·61-s + 63-s + 13·67-s − 2·69-s − 13·71-s − 8·73-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 1.21·17-s − 1.83·19-s + 0.436·21-s − 0.208·23-s − 0.769·27-s − 0.928·29-s + 0.898·31-s − 1.15·37-s − 0.640·39-s − 1.09·41-s + 0.609·43-s − 0.291·47-s − 6/7·49-s + 1.40·51-s + 0.137·53-s − 2.11·57-s − 0.390·59-s − 0.768·61-s + 0.125·63-s + 1.58·67-s − 0.240·69-s − 1.54·71-s − 0.936·73-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: $\chi_{9200} (1, \cdot )$
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 \( 1 - 2 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−7.62106484929794115692875680749, −6.82775449768462690281670327514, −6.03748053132603860162153792150, −5.25182454580537683307355844068, −4.46771541832328036978163887644, −3.71019004702725328029674046681, −3.03403677912797164355815174430, −2.21946699662204606454507005430, −1.56029224758525369379953605962, 0, 1.56029224758525369379953605962, 2.21946699662204606454507005430, 3.03403677912797164355815174430, 3.71019004702725328029674046681, 4.46771541832328036978163887644, 5.25182454580537683307355844068, 6.03748053132603860162153792150, 6.82775449768462690281670327514, 7.62106484929794115692875680749

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