Properties

Label 2-9800-1.1-c1-0-5
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  − 0.705·3-s − 2.50·9-s − 4.50·11-s − 5.09·13-s − 2·17-s − 3.09·19-s − 5.79·23-s + 3.88·27-s + 9.50·29-s + 5.41·31-s + 3.17·33-s − 7.09·37-s + 3.59·39-s − 6.59·41-s − 4.70·43-s − 10.0·47-s + 1.41·51-s − 9.91·53-s + 2.18·57-s + 8·59-s − 8.91·61-s + 0.117·67-s + 4.09·69-s − 8.18·73-s − 14.1·79-s + 4.76·81-s + 10.7·83-s + ⋯
L(s)  = 1  − 0.407·3-s − 0.833·9-s − 1.35·11-s − 1.41·13-s − 0.485·17-s − 0.709·19-s − 1.20·23-s + 0.747·27-s + 1.76·29-s + 0.971·31-s + 0.553·33-s − 1.16·37-s + 0.575·39-s − 1.02·41-s − 0.717·43-s − 1.47·47-s + 0.197·51-s − 1.36·53-s + 0.288·57-s + 1.04·59-s − 1.14·61-s + 0.0143·67-s + 0.492·69-s − 0.957·73-s − 1.59·79-s + 0.529·81-s + 1.17·83-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: 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

\(L(1)\) \(\approx\) \(0.2674376536\)
\(L(\frac12)\) \(\approx\) \(0.2674376536\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.705T + 3T^{2} \)
11 \( 1 + 4.50T + 11T^{2} \)
13 \( 1 + 5.09T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 3.09T + 19T^{2} \)
23 \( 1 + 5.79T + 23T^{2} \)
29 \( 1 - 9.50T + 29T^{2} \)
31 \( 1 - 5.41T + 31T^{2} \)
37 \( 1 + 7.09T + 37T^{2} \)
41 \( 1 + 6.59T + 41T^{2} \)
43 \( 1 + 4.70T + 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + 9.91T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + 8.91T + 61T^{2} \)
67 \( 1 - 0.117T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 8.18T + 73T^{2} \)
79 \( 1 + 14.1T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 - 4.09T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−7.86068607106263313650495422309, −6.79987888522979631825249498767, −6.40828839799576500647650957395, −5.53214096904552982464867162563, −4.89317413861182099315269531892, −4.52306765862731935660300389227, −3.18739309050141279171177749702, −2.65143889871719567822435566408, −1.85787885630984362399323552514, −0.23214942092026475231784630549, 0.23214942092026475231784630549, 1.85787885630984362399323552514, 2.65143889871719567822435566408, 3.18739309050141279171177749702, 4.52306765862731935660300389227, 4.89317413861182099315269531892, 5.53214096904552982464867162563, 6.40828839799576500647650957395, 6.79987888522979631825249498767, 7.86068607106263313650495422309

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