Properties

Label 2-9800-1.1-c1-0-89
Degree $2$
Conductor $9800$
Sign $-1$
Analytic cond. $78.2533$
Root an. cond. $8.84609$
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  − 2·3-s + 9-s − 4·11-s − 4·13-s + 4·19-s − 2·23-s + 4·27-s + 2·29-s + 8·33-s + 4·37-s + 8·39-s − 2·41-s − 6·43-s + 6·47-s − 4·53-s − 8·57-s + 12·59-s + 10·61-s + 14·67-s + 4·69-s + 8·71-s − 8·73-s + 16·79-s − 11·81-s − 2·83-s − 4·87-s − 6·89-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 1.20·11-s − 1.10·13-s + 0.917·19-s − 0.417·23-s + 0.769·27-s + 0.371·29-s + 1.39·33-s + 0.657·37-s + 1.28·39-s − 0.312·41-s − 0.914·43-s + 0.875·47-s − 0.549·53-s − 1.05·57-s + 1.56·59-s + 1.28·61-s + 1.71·67-s + 0.481·69-s + 0.949·71-s − 0.936·73-s + 1.80·79-s − 1.22·81-s − 0.219·83-s − 0.428·87-s − 0.635·89-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: yes
Arithmetic: yes
Character: $\chi_{9800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−7.14921286604979326658019660163, −6.70704043705589798939692431076, −5.77444390103934092747591743437, −5.25454667818013035065281812785, −4.92094480034658234190771257579, −3.94230043837437753170237232424, −2.88727452627385173253869196408, −2.24846064180628551899679631269, −0.903711169250688522174204642296, 0, 0.903711169250688522174204642296, 2.24846064180628551899679631269, 2.88727452627385173253869196408, 3.94230043837437753170237232424, 4.92094480034658234190771257579, 5.25454667818013035065281812785, 5.77444390103934092747591743437, 6.70704043705589798939692431076, 7.14921286604979326658019660163

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