Properties

Label 2-9800-1.1-c1-0-85
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  + 2.18·3-s + 1.76·9-s + 3.59·11-s + 5.85·13-s − 6.36·17-s − 4.50·19-s − 2.62·23-s − 2.68·27-s + 2.05·29-s + 6.62·31-s + 7.85·33-s + 5.91·37-s + 12.7·39-s − 7.22·41-s + 5.34·43-s + 2.31·47-s − 13.8·51-s + 4.10·53-s − 9.83·57-s + 5.07·59-s + 7.37·61-s − 3.39·67-s − 5.73·69-s + 16.7·71-s + 14.7·73-s − 3.25·79-s − 11.1·81-s + ⋯
L(s)  = 1  + 1.26·3-s + 0.589·9-s + 1.08·11-s + 1.62·13-s − 1.54·17-s − 1.03·19-s − 0.548·23-s − 0.517·27-s + 0.382·29-s + 1.19·31-s + 1.36·33-s + 0.972·37-s + 2.04·39-s − 1.12·41-s + 0.815·43-s + 0.338·47-s − 1.94·51-s + 0.564·53-s − 1.30·57-s + 0.660·59-s + 0.944·61-s − 0.414·67-s − 0.690·69-s + 1.98·71-s + 1.72·73-s − 0.366·79-s − 1.24·81-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: $\chi_{9800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.749737977\)
\(L(\frac12)\) \(\approx\) \(3.749737977\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.18T + 3T^{2} \)
11 \( 1 - 3.59T + 11T^{2} \)
13 \( 1 - 5.85T + 13T^{2} \)
17 \( 1 + 6.36T + 17T^{2} \)
19 \( 1 + 4.50T + 19T^{2} \)
23 \( 1 + 2.62T + 23T^{2} \)
29 \( 1 - 2.05T + 29T^{2} \)
31 \( 1 - 6.62T + 31T^{2} \)
37 \( 1 - 5.91T + 37T^{2} \)
41 \( 1 + 7.22T + 41T^{2} \)
43 \( 1 - 5.34T + 43T^{2} \)
47 \( 1 - 2.31T + 47T^{2} \)
53 \( 1 - 4.10T + 53T^{2} \)
59 \( 1 - 5.07T + 59T^{2} \)
61 \( 1 - 7.37T + 61T^{2} \)
67 \( 1 + 3.39T + 67T^{2} \)
71 \( 1 - 16.7T + 71T^{2} \)
73 \( 1 - 14.7T + 73T^{2} \)
79 \( 1 + 3.25T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 - 17.1T + 97T^{2} \)
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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

−8.019909231422216524108637009967, −6.82541780355671644135178090272, −6.50472664056395981600028923211, −5.83292682858036977997341898376, −4.58708868894098433818637247089, −3.97905034004753664663550584909, −3.56237860341802327964671384362, −2.49662122102152230702388645939, −1.96129716865278841929589310149, −0.879651669354760535680342041249, 0.879651669354760535680342041249, 1.96129716865278841929589310149, 2.49662122102152230702388645939, 3.56237860341802327964671384362, 3.97905034004753664663550584909, 4.58708868894098433818637247089, 5.83292682858036977997341898376, 6.50472664056395981600028923211, 6.82541780355671644135178090272, 8.019909231422216524108637009967

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