Properties

Label 2-9800-1.1-c1-0-23
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  − 1.87·3-s + 0.534·9-s − 3.29·11-s + 4.19·13-s − 1.43·17-s − 1.24·19-s + 0.272·23-s + 4.63·27-s − 2.36·29-s + 3.72·31-s + 6.19·33-s − 0.169·37-s − 7.88·39-s − 11.6·41-s + 10.1·43-s + 3.12·47-s + 2.70·51-s − 9.24·53-s + 2.33·57-s − 9.07·59-s − 7.27·61-s + 13.1·67-s − 0.512·69-s + 6.87·71-s − 15.2·73-s + 14.9·79-s − 10.3·81-s + ⋯
L(s)  = 1  − 1.08·3-s + 0.178·9-s − 0.993·11-s + 1.16·13-s − 0.348·17-s − 0.285·19-s + 0.0568·23-s + 0.892·27-s − 0.438·29-s + 0.669·31-s + 1.07·33-s − 0.0279·37-s − 1.26·39-s − 1.82·41-s + 1.54·43-s + 0.455·47-s + 0.378·51-s − 1.27·53-s + 0.309·57-s − 1.18·59-s − 0.930·61-s + 1.60·67-s − 0.0617·69-s + 0.815·71-s − 1.78·73-s + 1.68·79-s − 1.14·81-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: \(0\)
Selberg data: \((2,\ 9800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8639647760\)
\(L(\frac12)\) \(\approx\) \(0.8639647760\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.87T + 3T^{2} \)
11 \( 1 + 3.29T + 11T^{2} \)
13 \( 1 - 4.19T + 13T^{2} \)
17 \( 1 + 1.43T + 17T^{2} \)
19 \( 1 + 1.24T + 19T^{2} \)
23 \( 1 - 0.272T + 23T^{2} \)
29 \( 1 + 2.36T + 29T^{2} \)
31 \( 1 - 3.72T + 31T^{2} \)
37 \( 1 + 0.169T + 37T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 - 3.12T + 47T^{2} \)
53 \( 1 + 9.24T + 53T^{2} \)
59 \( 1 + 9.07T + 59T^{2} \)
61 \( 1 + 7.27T + 61T^{2} \)
67 \( 1 - 13.1T + 67T^{2} \)
71 \( 1 - 6.87T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 - 0.167T + 83T^{2} \)
89 \( 1 + 3.09T + 89T^{2} \)
97 \( 1 - 6.60T + 97T^{2} \)
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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

−7.68416037909433967446169967775, −6.75620724639113738722662407012, −6.25385851608382725211741053856, −5.66360450768308837966014595427, −5.04198056040237010434791307072, −4.37576923104059885573756165380, −3.44190203285991599167320536647, −2.61826122248111629899356513431, −1.54513486807675207950574556731, −0.47598627471980384270154156504, 0.47598627471980384270154156504, 1.54513486807675207950574556731, 2.61826122248111629899356513431, 3.44190203285991599167320536647, 4.37576923104059885573756165380, 5.04198056040237010434791307072, 5.66360450768308837966014595427, 6.25385851608382725211741053856, 6.75620724639113738722662407012, 7.68416037909433967446169967775

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