Properties

Label 2-9200-1.1-c1-0-186
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 − 3·11-s − 13-s + 19-s + 2·21-s − 23-s − 4·27-s − 3·29-s − 2·31-s − 6·33-s + 2·37-s − 2·39-s + 3·41-s + 43-s − 6·49-s − 12·53-s + 2·57-s + 6·59-s + 2·61-s + 63-s − 8·67-s − 2·69-s + 6·71-s − 7·73-s − 3·77-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s + 0.229·19-s + 0.436·21-s − 0.208·23-s − 0.769·27-s − 0.557·29-s − 0.359·31-s − 1.04·33-s + 0.328·37-s − 0.320·39-s + 0.468·41-s + 0.152·43-s − 6/7·49-s − 1.64·53-s + 0.264·57-s + 0.781·59-s + 0.256·61-s + 0.125·63-s − 0.977·67-s − 0.240·69-s + 0.712·71-s − 0.819·73-s − 0.341·77-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 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−7.65002721082632609200722036671, −6.91455280588508681626939281594, −5.92797838968929404789038515173, −5.28697834690244400556048162104, −4.51368047706232537231115928321, −3.69746724152175177080603873962, −2.94060991841787652180730775014, −2.34746491892901115419717792287, −1.50001760183063569158676053738, 0, 1.50001760183063569158676053738, 2.34746491892901115419717792287, 2.94060991841787652180730775014, 3.69746724152175177080603873962, 4.51368047706232537231115928321, 5.28697834690244400556048162104, 5.92797838968929404789038515173, 6.91455280588508681626939281594, 7.65002721082632609200722036671

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