Properties

Label 2-9800-1.1-c1-0-119
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  − 3-s − 2·9-s − 5·11-s + 13-s + 3·17-s + 6·19-s + 6·23-s + 5·27-s − 9·29-s + 5·33-s − 6·37-s − 39-s − 8·41-s − 6·43-s + 3·47-s − 3·51-s + 12·53-s − 6·57-s − 8·59-s + 4·61-s + 4·67-s − 6·69-s + 8·71-s + 10·73-s − 3·79-s + 81-s − 12·83-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s − 1.50·11-s + 0.277·13-s + 0.727·17-s + 1.37·19-s + 1.25·23-s + 0.962·27-s − 1.67·29-s + 0.870·33-s − 0.986·37-s − 0.160·39-s − 1.24·41-s − 0.914·43-s + 0.437·47-s − 0.420·51-s + 1.64·53-s − 0.794·57-s − 1.04·59-s + 0.512·61-s + 0.488·67-s − 0.722·69-s + 0.949·71-s + 1.17·73-s − 0.337·79-s + 1/9·81-s − 1.31·83-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 \( 1 + T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−7.35506837323567972485067911086, −6.68703256181804606281147125839, −5.69124619217865701050329595606, −5.29384395404242793953990195152, −4.98809239944162573811126658586, −3.58039186109375468513509715851, −3.14114445601193552512668015535, −2.21471885119434180517790562935, −1.03834075997729573077765602229, 0, 1.03834075997729573077765602229, 2.21471885119434180517790562935, 3.14114445601193552512668015535, 3.58039186109375468513509715851, 4.98809239944162573811126658586, 5.29384395404242793953990195152, 5.69124619217865701050329595606, 6.68703256181804606281147125839, 7.35506837323567972485067911086

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