Properties

Label 2-9800-1.1-c1-0-16
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  − 0.546·3-s − 2.70·9-s − 1.70·11-s + 0.546·13-s − 3.11·17-s − 4.75·19-s − 5.40·23-s + 3.11·27-s − 0.298·29-s + 7.32·31-s + 0.929·33-s − 5.40·37-s − 0.298·39-s − 7.32·41-s + 4·43-s − 8.25·47-s + 1.70·51-s − 1.40·53-s + 2.59·57-s − 7.70·59-s + 9.89·61-s + 8·67-s + 2.95·69-s − 5.40·71-s + 11.3·73-s + 11.1·79-s + 6.40·81-s + ⋯
L(s)  = 1  − 0.315·3-s − 0.900·9-s − 0.513·11-s + 0.151·13-s − 0.755·17-s − 1.09·19-s − 1.12·23-s + 0.599·27-s − 0.0554·29-s + 1.31·31-s + 0.161·33-s − 0.888·37-s − 0.0477·39-s − 1.14·41-s + 0.609·43-s − 1.20·47-s + 0.238·51-s − 0.192·53-s + 0.343·57-s − 1.00·59-s + 1.26·61-s + 0.977·67-s + 0.355·69-s − 0.641·71-s + 1.33·73-s + 1.24·79-s + 0.711·81-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: \(0\)
Selberg data: \((2,\ 9800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7964687241\)
\(L(\frac12)\) \(\approx\) \(0.7964687241\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.546T + 3T^{2} \)
11 \( 1 + 1.70T + 11T^{2} \)
13 \( 1 - 0.546T + 13T^{2} \)
17 \( 1 + 3.11T + 17T^{2} \)
19 \( 1 + 4.75T + 19T^{2} \)
23 \( 1 + 5.40T + 23T^{2} \)
29 \( 1 + 0.298T + 29T^{2} \)
31 \( 1 - 7.32T + 31T^{2} \)
37 \( 1 + 5.40T + 37T^{2} \)
41 \( 1 + 7.32T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 8.25T + 47T^{2} \)
53 \( 1 + 1.40T + 53T^{2} \)
59 \( 1 + 7.70T + 59T^{2} \)
61 \( 1 - 9.89T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 5.40T + 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 + 9.89T + 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 + 5.29T + 97T^{2} \)
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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

−7.80464314329343738987091055915, −6.74412395629802883828062070052, −6.38567730948970172874444761523, −5.65155653337521157677984721116, −4.95711658494598624678947483070, −4.27541290398350607731316516666, −3.39797257562260009540463345476, −2.54711345054479344072743172033, −1.83225397888700437354988471672, −0.40876501115478679853643355504, 0.40876501115478679853643355504, 1.83225397888700437354988471672, 2.54711345054479344072743172033, 3.39797257562260009540463345476, 4.27541290398350607731316516666, 4.95711658494598624678947483070, 5.65155653337521157677984721116, 6.38567730948970172874444761523, 6.74412395629802883828062070052, 7.80464314329343738987091055915

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