Properties

Label 2-9800-1.1-c1-0-128
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 + 2·11-s − 4·17-s − 2·19-s − 23-s + 5·27-s + 9·29-s + 4·31-s − 2·33-s − 4·37-s + 41-s − 9·43-s + 4·51-s + 10·53-s + 2·57-s − 10·59-s + 9·61-s − 5·67-s + 69-s + 14·71-s − 12·73-s + 14·79-s + 81-s − 11·83-s − 9·87-s − 15·89-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s + 0.603·11-s − 0.970·17-s − 0.458·19-s − 0.208·23-s + 0.962·27-s + 1.67·29-s + 0.718·31-s − 0.348·33-s − 0.657·37-s + 0.156·41-s − 1.37·43-s + 0.560·51-s + 1.37·53-s + 0.264·57-s − 1.30·59-s + 1.15·61-s − 0.610·67-s + 0.120·69-s + 1.66·71-s − 1.40·73-s + 1.57·79-s + 1/9·81-s − 1.20·83-s − 0.964·87-s − 1.58·89-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: $\chi_{9800} (1, \cdot )$
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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 9 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−7.11940382916562769380244118564, −6.49064321914241615745315723343, −6.17161502867647856199519252884, −5.21839308674570266688020793945, −4.66776323393633705224746462061, −3.90395022502470222098843188524, −2.97111346358414364452819966931, −2.20137090646183567088631138610, −1.07748740319160806845883403538, 0, 1.07748740319160806845883403538, 2.20137090646183567088631138610, 2.97111346358414364452819966931, 3.90395022502470222098843188524, 4.66776323393633705224746462061, 5.21839308674570266688020793945, 6.17161502867647856199519252884, 6.49064321914241615745315723343, 7.11940382916562769380244118564

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