Properties

Label 2-9200-1.1-c1-0-80
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  − 3·3-s − 4·7-s + 6·9-s − 2·11-s + 5·13-s − 4·17-s + 2·19-s + 12·21-s + 23-s − 9·27-s − 7·29-s + 3·31-s + 6·33-s − 2·37-s − 15·39-s − 9·41-s − 8·43-s + 9·47-s + 9·49-s + 12·51-s − 2·53-s − 6·57-s − 2·61-s − 24·63-s + 14·67-s − 3·69-s + 3·71-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.51·7-s + 2·9-s − 0.603·11-s + 1.38·13-s − 0.970·17-s + 0.458·19-s + 2.61·21-s + 0.208·23-s − 1.73·27-s − 1.29·29-s + 0.538·31-s + 1.04·33-s − 0.328·37-s − 2.40·39-s − 1.40·41-s − 1.21·43-s + 1.31·47-s + 9/7·49-s + 1.68·51-s − 0.274·53-s − 0.794·57-s − 0.256·61-s − 3.02·63-s + 1.71·67-s − 0.361·69-s + 0.356·71-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 \( 1 + p T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
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 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−6.90035333980951981978937222896, −6.66594669567581980430729862767, −6.02771573886787652498895446151, −5.47556142437483889479834900997, −4.81024105608428314617771606455, −3.83476933689227639877998666629, −3.29336370154811067963272498287, −2.01206047983172759500763124469, −0.832565523311458848680696537063, 0, 0.832565523311458848680696537063, 2.01206047983172759500763124469, 3.29336370154811067963272498287, 3.83476933689227639877998666629, 4.81024105608428314617771606455, 5.47556142437483889479834900997, 6.02771573886787652498895446151, 6.66594669567581980430729862767, 6.90035333980951981978937222896

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