Properties

Label 2-9800-1.1-c1-0-175
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.41·3-s + 2.82·9-s − 4.82·11-s − 2·13-s − 3.65·17-s + 5.65·19-s + 8.41·23-s − 0.414·27-s − 2.17·29-s − 4.82·31-s − 11.6·33-s + 5.65·37-s − 4.82·39-s + 0.171·41-s − 12.8·43-s − 0.343·47-s − 8.82·51-s − 5.65·53-s + 13.6·57-s − 4·59-s − 4.65·61-s − 6.89·67-s + 20.3·69-s − 12·71-s + 7.65·73-s − 4·79-s − 9.48·81-s + ⋯
L(s)  = 1  + 1.39·3-s + 0.942·9-s − 1.45·11-s − 0.554·13-s − 0.886·17-s + 1.29·19-s + 1.75·23-s − 0.0797·27-s − 0.403·29-s − 0.867·31-s − 2.02·33-s + 0.929·37-s − 0.773·39-s + 0.0267·41-s − 1.96·43-s − 0.0500·47-s − 1.23·51-s − 0.777·53-s + 1.80·57-s − 0.520·59-s − 0.596·61-s − 0.842·67-s + 2.44·69-s − 1.42·71-s + 0.896·73-s − 0.450·79-s − 1.05·81-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: $\chi_{9800} (1, \cdot )$
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 \( 1 - 2.41T + 3T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 3.65T + 17T^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
23 \( 1 - 8.41T + 23T^{2} \)
29 \( 1 + 2.17T + 29T^{2} \)
31 \( 1 + 4.82T + 31T^{2} \)
37 \( 1 - 5.65T + 37T^{2} \)
41 \( 1 - 0.171T + 41T^{2} \)
43 \( 1 + 12.8T + 43T^{2} \)
47 \( 1 + 0.343T + 47T^{2} \)
53 \( 1 + 5.65T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 4.65T + 61T^{2} \)
67 \( 1 + 6.89T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 7.65T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−7.36704910371947464701627265874, −7.07526705545864904323204606151, −5.91118536937455914598724625034, −5.07798928232015481359769948541, −4.63910898773724583214304905114, −3.46894692130880219024768769752, −2.98188273920396512757567966598, −2.41389015489771500166816105683, −1.48617882024117632317609111240, 0, 1.48617882024117632317609111240, 2.41389015489771500166816105683, 2.98188273920396512757567966598, 3.46894692130880219024768769752, 4.63910898773724583214304905114, 5.07798928232015481359769948541, 5.91118536937455914598724625034, 7.07526705545864904323204606151, 7.36704910371947464701627265874

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