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Properties

Label 2-9800-1.1-c1-0-141
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  − 2·3-s + 9-s + 4·11-s + 2·13-s + 3·17-s + 3·23-s + 4·27-s − 6·29-s − 9·31-s − 8·33-s − 4·39-s − 5·41-s − 6·43-s − 9·47-s − 6·51-s − 6·53-s − 8·59-s − 8·61-s + 14·67-s − 6·69-s + 11·71-s + 2·73-s + 9·79-s − 11·81-s + 6·83-s + 12·87-s − 11·89-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.727·17-s + 0.625·23-s + 0.769·27-s − 1.11·29-s − 1.61·31-s − 1.39·33-s − 0.640·39-s − 0.780·41-s − 0.914·43-s − 1.31·47-s − 0.840·51-s − 0.824·53-s − 1.04·59-s − 1.02·61-s + 1.71·67-s − 0.722·69-s + 1.30·71-s + 0.234·73-s + 1.01·79-s − 1.22·81-s + 0.658·83-s + 1.28·87-s − 1.16·89-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: 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 11 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−7.11044115657832784025917734940, −6.53044058171096431969337711478, −6.02294350817926068884566442199, −5.29081621388565372134995285742, −4.80573565140868679309569618374, −3.70913989162551534996288906458, −3.32847973219145791341358740365, −1.86197584730658228901550371371, −1.13880264951000867601933654827, 0, 1.13880264951000867601933654827, 1.86197584730658228901550371371, 3.32847973219145791341358740365, 3.70913989162551534996288906458, 4.80573565140868679309569618374, 5.29081621388565372134995285742, 6.02294350817926068884566442199, 6.53044058171096431969337711478, 7.11044115657832784025917734940

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