Properties

Label 2-9800-1.1-c1-0-19
Degree $2$
Conductor $9800$
Sign $1$
Analytic cond. $78.2533$
Root an. cond. $8.84609$
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  − 0.879·3-s − 2.22·9-s − 3.18·11-s + 0.490·13-s − 0.958·17-s − 2.10·19-s + 6.29·23-s + 4.59·27-s + 0.588·29-s − 7.47·31-s + 2.80·33-s − 1.50·37-s − 0.431·39-s − 4.29·41-s − 5.92·43-s + 5.16·47-s + 0.842·51-s − 13.1·53-s + 1.85·57-s + 4.63·59-s + 13.9·61-s − 7.27·67-s − 5.53·69-s + 12.1·71-s − 10.1·73-s − 2.53·79-s + 2.63·81-s + ⋯
L(s)  = 1  − 0.507·3-s − 0.742·9-s − 0.960·11-s + 0.135·13-s − 0.232·17-s − 0.483·19-s + 1.31·23-s + 0.884·27-s + 0.109·29-s − 1.34·31-s + 0.487·33-s − 0.248·37-s − 0.0690·39-s − 0.670·41-s − 0.902·43-s + 0.753·47-s + 0.117·51-s − 1.81·53-s + 0.245·57-s + 0.603·59-s + 1.78·61-s − 0.888·67-s − 0.665·69-s + 1.44·71-s − 1.18·73-s − 0.284·79-s + 0.293·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9800\)    =    \(2^{3} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(78.2533\)
Root analytic conductor: \(8.84609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8635614593\)
\(L(\frac12)\) \(\approx\) \(0.8635614593\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3 \( 1 + 0.879T + 3T^{2} \)
11 \( 1 + 3.18T + 11T^{2} \)
13 \( 1 - 0.490T + 13T^{2} \)
17 \( 1 + 0.958T + 17T^{2} \)
19 \( 1 + 2.10T + 19T^{2} \)
23 \( 1 - 6.29T + 23T^{2} \)
29 \( 1 - 0.588T + 29T^{2} \)
31 \( 1 + 7.47T + 31T^{2} \)
37 \( 1 + 1.50T + 37T^{2} \)
41 \( 1 + 4.29T + 41T^{2} \)
43 \( 1 + 5.92T + 43T^{2} \)
47 \( 1 - 5.16T + 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 - 4.63T + 59T^{2} \)
61 \( 1 - 13.9T + 61T^{2} \)
67 \( 1 + 7.27T + 67T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 + 2.53T + 79T^{2} \)
83 \( 1 + 4.80T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−7.61791825365977310794386223834, −6.91414066254040351260500308275, −6.33336489406684406825413109084, −5.37088733249355273377345641895, −5.24645848441735856054898800782, −4.28681075647765287827531930184, −3.30380290991678808184116635229, −2.68501371170429341434972835815, −1.70538679090166193674771108843, −0.44076388390519203488949630887, 0.44076388390519203488949630887, 1.70538679090166193674771108843, 2.68501371170429341434972835815, 3.30380290991678808184116635229, 4.28681075647765287827531930184, 5.24645848441735856054898800782, 5.37088733249355273377345641895, 6.33336489406684406825413109084, 6.91414066254040351260500308275, 7.61791825365977310794386223834

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