Properties

Label 2-9800-1.1-c1-0-183
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  + 2.28·3-s + 2.23·9-s − 11-s + 0.874·13-s − 4.57·17-s + 3.16·19-s − 23-s − 1.74·27-s − 8.70·29-s − 2.62·31-s − 2.28·33-s − 0.236·37-s + 2·39-s − 9.69·41-s + 2.23·43-s + 1.41·47-s − 10.4·51-s + 1.23·53-s + 7.23·57-s + 11.1·59-s − 5.11·61-s − 7.47·67-s − 2.28·69-s + 6.70·71-s − 2.95·73-s − 11.9·79-s − 10.7·81-s + ⋯
L(s)  = 1  + 1.32·3-s + 0.745·9-s − 0.301·11-s + 0.242·13-s − 1.10·17-s + 0.725·19-s − 0.208·23-s − 0.336·27-s − 1.61·29-s − 0.470·31-s − 0.398·33-s − 0.0388·37-s + 0.320·39-s − 1.51·41-s + 0.340·43-s + 0.206·47-s − 1.46·51-s + 0.169·53-s + 0.958·57-s + 1.44·59-s − 0.655·61-s − 0.912·67-s − 0.275·69-s + 0.796·71-s − 0.345·73-s − 1.34·79-s − 1.18·81-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: \(1\)
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 \( 1 - 2.28T + 3T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 - 0.874T + 13T^{2} \)
17 \( 1 + 4.57T + 17T^{2} \)
19 \( 1 - 3.16T + 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + 8.70T + 29T^{2} \)
31 \( 1 + 2.62T + 31T^{2} \)
37 \( 1 + 0.236T + 37T^{2} \)
41 \( 1 + 9.69T + 41T^{2} \)
43 \( 1 - 2.23T + 43T^{2} \)
47 \( 1 - 1.41T + 47T^{2} \)
53 \( 1 - 1.23T + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 + 5.11T + 61T^{2} \)
67 \( 1 + 7.47T + 67T^{2} \)
71 \( 1 - 6.70T + 71T^{2} \)
73 \( 1 + 2.95T + 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 + 8.27T + 83T^{2} \)
89 \( 1 + 1.95T + 89T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−7.33521975722468003598846475994, −6.97758162472285137808347491905, −5.90816006475011874423450457578, −5.30252674524540328930652526651, −4.31797664535834713360278932306, −3.69438448391207074587248076696, −3.01331907287267966781612295737, −2.24431956421536731978367102176, −1.55031827529014905477045036704, 0, 1.55031827529014905477045036704, 2.24431956421536731978367102176, 3.01331907287267966781612295737, 3.69438448391207074587248076696, 4.31797664535834713360278932306, 5.30252674524540328930652526651, 5.90816006475011874423450457578, 6.97758162472285137808347491905, 7.33521975722468003598846475994

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