Properties

Label 2-9200-1.1-c1-0-8
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  + 0.356·3-s − 2.46·7-s − 2.87·9-s − 1.61·11-s − 2.62·13-s − 2.58·17-s − 4.02·19-s − 0.878·21-s − 23-s − 2.09·27-s − 7.08·29-s + 4.58·31-s − 0.575·33-s + 2.96·37-s − 0.935·39-s − 5.71·41-s + 2.30·43-s − 6.88·47-s − 0.929·49-s − 0.922·51-s − 6.76·53-s − 1.43·57-s − 2.53·59-s + 9.25·61-s + 7.07·63-s + 15.7·67-s − 0.356·69-s + ⋯
L(s)  = 1  + 0.205·3-s − 0.931·7-s − 0.957·9-s − 0.487·11-s − 0.728·13-s − 0.627·17-s − 0.923·19-s − 0.191·21-s − 0.208·23-s − 0.402·27-s − 1.31·29-s + 0.823·31-s − 0.100·33-s + 0.486·37-s − 0.149·39-s − 0.891·41-s + 0.351·43-s − 1.00·47-s − 0.132·49-s − 0.129·51-s − 0.929·53-s − 0.190·57-s − 0.329·59-s + 1.18·61-s + 0.891·63-s + 1.92·67-s − 0.0429·69-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

\(L(1)\) \(\approx\) \(0.6172286630\)
\(L(\frac12)\) \(\approx\) \(0.6172286630\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.356T + 3T^{2} \)
7 \( 1 + 2.46T + 7T^{2} \)
11 \( 1 + 1.61T + 11T^{2} \)
13 \( 1 + 2.62T + 13T^{2} \)
17 \( 1 + 2.58T + 17T^{2} \)
19 \( 1 + 4.02T + 19T^{2} \)
29 \( 1 + 7.08T + 29T^{2} \)
31 \( 1 - 4.58T + 31T^{2} \)
37 \( 1 - 2.96T + 37T^{2} \)
41 \( 1 + 5.71T + 41T^{2} \)
43 \( 1 - 2.30T + 43T^{2} \)
47 \( 1 + 6.88T + 47T^{2} \)
53 \( 1 + 6.76T + 53T^{2} \)
59 \( 1 + 2.53T + 59T^{2} \)
61 \( 1 - 9.25T + 61T^{2} \)
67 \( 1 - 15.7T + 67T^{2} \)
71 \( 1 - 5.25T + 71T^{2} \)
73 \( 1 - 6.03T + 73T^{2} \)
79 \( 1 - 1.35T + 79T^{2} \)
83 \( 1 - 8.59T + 83T^{2} \)
89 \( 1 + 8.84T + 89T^{2} \)
97 \( 1 - 8.01T + 97T^{2} \)
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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

−8.029475549395401158639430516964, −6.78858227680266472115235203028, −6.54818800027464073671877364249, −5.64481267418146345703176331570, −5.05034558937650329583472762321, −4.13400006963888148369406019498, −3.35469692110891743328630784328, −2.62526793159429478120404861540, −2.00086533563197134091058981898, −0.34402813475140988443589186584, 0.34402813475140988443589186584, 2.00086533563197134091058981898, 2.62526793159429478120404861540, 3.35469692110891743328630784328, 4.13400006963888148369406019498, 5.05034558937650329583472762321, 5.64481267418146345703176331570, 6.54818800027464073671877364249, 6.78858227680266472115235203028, 8.029475549395401158639430516964

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