Properties

Label 2-9800-1.1-c1-0-127
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  + 2.87·3-s + 5.29·9-s + 1.46·11-s + 2.22·13-s + 7.26·17-s + 5.48·19-s − 2.51·23-s + 6.60·27-s − 7.12·29-s + 6.51·31-s + 4.22·33-s − 6.90·37-s + 6.39·39-s + 11.3·41-s − 3.31·43-s − 8.36·47-s + 20.9·51-s + 7.00·53-s + 15.8·57-s − 9.07·59-s − 11.2·61-s + 6.41·67-s − 7.24·69-s − 10.5·71-s + 10.5·73-s + 10.1·79-s + 3.14·81-s + ⋯
L(s)  = 1  + 1.66·3-s + 1.76·9-s + 0.441·11-s + 0.616·13-s + 1.76·17-s + 1.25·19-s − 0.524·23-s + 1.27·27-s − 1.32·29-s + 1.17·31-s + 0.734·33-s − 1.13·37-s + 1.02·39-s + 1.76·41-s − 0.505·43-s − 1.22·47-s + 2.93·51-s + 0.962·53-s + 2.09·57-s − 1.18·59-s − 1.43·61-s + 0.783·67-s − 0.872·69-s − 1.24·71-s + 1.23·73-s + 1.14·79-s + 0.349·81-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

\(L(1)\) \(\approx\) \(4.973234625\)
\(L(\frac12)\) \(\approx\) \(4.973234625\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.87T + 3T^{2} \)
11 \( 1 - 1.46T + 11T^{2} \)
13 \( 1 - 2.22T + 13T^{2} \)
17 \( 1 - 7.26T + 17T^{2} \)
19 \( 1 - 5.48T + 19T^{2} \)
23 \( 1 + 2.51T + 23T^{2} \)
29 \( 1 + 7.12T + 29T^{2} \)
31 \( 1 - 6.51T + 31T^{2} \)
37 \( 1 + 6.90T + 37T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 + 3.31T + 43T^{2} \)
47 \( 1 + 8.36T + 47T^{2} \)
53 \( 1 - 7.00T + 53T^{2} \)
59 \( 1 + 9.07T + 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 - 6.41T + 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 - 16.4T + 83T^{2} \)
89 \( 1 + 9.83T + 89T^{2} \)
97 \( 1 + 2.09T + 97T^{2} \)
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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

−7.77463823285330093491273387566, −7.35553067710443221843139609382, −6.38450749766749511785576260731, −5.64320290098593803310922892562, −4.79561806074447157559884750474, −3.73477665348731309023823373982, −3.49325794717638403380155091369, −2.76350737635017553950922573916, −1.76498340255698093387370757446, −1.07140873516174162550983641302, 1.07140873516174162550983641302, 1.76498340255698093387370757446, 2.76350737635017553950922573916, 3.49325794717638403380155091369, 3.73477665348731309023823373982, 4.79561806074447157559884750474, 5.64320290098593803310922892562, 6.38450749766749511785576260731, 7.35553067710443221843139609382, 7.77463823285330093491273387566

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