Properties

Label 2-9200-1.1-c1-0-145
Degree $2$
Conductor $9200$
Sign $1$
Analytic cond. $73.4623$
Root an. cond. $8.57101$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.56·3-s + 5.12·7-s + 3.56·9-s + 4·11-s + 0.561·13-s + 3.12·17-s − 4·19-s + 13.1·21-s + 23-s + 1.43·27-s − 8.56·29-s − 1.43·31-s + 10.2·33-s + 7.12·37-s + 1.43·39-s + 0.561·41-s − 9.12·43-s − 3.68·47-s + 19.2·49-s + 8·51-s + 4.24·53-s − 10.2·57-s + 6.24·59-s + 11.1·61-s + 18.2·63-s + 6.24·67-s + 2.56·69-s + ⋯
L(s)  = 1  + 1.47·3-s + 1.93·7-s + 1.18·9-s + 1.20·11-s + 0.155·13-s + 0.757·17-s − 0.917·19-s + 2.86·21-s + 0.208·23-s + 0.276·27-s − 1.58·29-s − 0.258·31-s + 1.78·33-s + 1.17·37-s + 0.230·39-s + 0.0876·41-s − 1.39·43-s − 0.537·47-s + 2.74·49-s + 1.12·51-s + 0.583·53-s − 1.35·57-s + 0.813·59-s + 1.42·61-s + 2.29·63-s + 0.763·67-s + 0.308·69-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.427027477\)
\(L(\frac12)\) \(\approx\) \(5.427027477\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.56T + 3T^{2} \)
7 \( 1 - 5.12T + 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 0.561T + 13T^{2} \)
17 \( 1 - 3.12T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
29 \( 1 + 8.56T + 29T^{2} \)
31 \( 1 + 1.43T + 31T^{2} \)
37 \( 1 - 7.12T + 37T^{2} \)
41 \( 1 - 0.561T + 41T^{2} \)
43 \( 1 + 9.12T + 43T^{2} \)
47 \( 1 + 3.68T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 - 6.24T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 - 6.24T + 67T^{2} \)
71 \( 1 + 3.68T + 71T^{2} \)
73 \( 1 + 16.5T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−7.85134227635075918275125901416, −7.37344667194435769128953510571, −6.50449685657283727127598002076, −5.53620681204273916318659825016, −4.81415229100274953150224858338, −3.95184980699086291238857053161, −3.65544640242403380565184233164, −2.43493218215084872038371293093, −1.83946166624433489070707842368, −1.17558344795246918171055634894, 1.17558344795246918171055634894, 1.83946166624433489070707842368, 2.43493218215084872038371293093, 3.65544640242403380565184233164, 3.95184980699086291238857053161, 4.81415229100274953150224858338, 5.53620681204273916318659825016, 6.50449685657283727127598002076, 7.37344667194435769128953510571, 7.85134227635075918275125901416

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