Properties

Label 2-9200-1.1-c1-0-206
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·7-s − 2·9-s − 3·11-s − 2·13-s − 17-s + 19-s + 4·21-s − 23-s + 5·27-s − 8·31-s + 3·33-s − 2·37-s + 2·39-s + 41-s − 12·43-s − 6·47-s + 9·49-s + 51-s − 4·53-s − 57-s − 12·59-s + 8·63-s − 13·67-s + 69-s − 12·71-s − 17·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s − 2/3·9-s − 0.904·11-s − 0.554·13-s − 0.242·17-s + 0.229·19-s + 0.872·21-s − 0.208·23-s + 0.962·27-s − 1.43·31-s + 0.522·33-s − 0.328·37-s + 0.320·39-s + 0.156·41-s − 1.82·43-s − 0.875·47-s + 9/7·49-s + 0.140·51-s − 0.549·53-s − 0.132·57-s − 1.56·59-s + 1.00·63-s − 1.58·67-s + 0.120·69-s − 1.42·71-s − 1.98·73-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: \(2\)
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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 17 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
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

−6.90759058524442549243077267272, −6.33938767117391270875111272296, −5.66570666613880294167698985103, −5.16036445105604375764056352421, −4.27959983417669839519431767188, −3.10283302863770691398757003422, −2.98408256359862025939295963858, −1.73267118980449994702770580450, 0, 0, 1.73267118980449994702770580450, 2.98408256359862025939295963858, 3.10283302863770691398757003422, 4.27959983417669839519431767188, 5.16036445105604375764056352421, 5.66570666613880294167698985103, 6.33938767117391270875111272296, 6.90759058524442549243077267272

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