Properties

Label 2-9800-1.1-c1-0-63
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  + 1.18·3-s − 1.59·9-s + 0.230·11-s + 2.27·13-s − 6.53·17-s − 0.260·19-s + 8.87·23-s − 5.44·27-s + 5.42·29-s + 4.87·31-s + 0.272·33-s + 1.15·37-s + 2.69·39-s + 4.43·41-s − 4.17·43-s − 0.923·47-s − 7.73·51-s + 2.13·53-s − 0.308·57-s − 5.07·59-s + 8.88·61-s − 8.15·67-s + 10.5·69-s − 9.06·71-s − 6.00·73-s + 0.112·79-s − 1.65·81-s + ⋯
L(s)  = 1  + 0.683·3-s − 0.532·9-s + 0.0694·11-s + 0.630·13-s − 1.58·17-s − 0.0596·19-s + 1.84·23-s − 1.04·27-s + 1.00·29-s + 0.874·31-s + 0.0474·33-s + 0.189·37-s + 0.430·39-s + 0.692·41-s − 0.637·43-s − 0.134·47-s − 1.08·51-s + 0.293·53-s − 0.0408·57-s − 0.660·59-s + 1.13·61-s − 0.996·67-s + 1.26·69-s − 1.07·71-s − 0.703·73-s + 0.0127·79-s − 0.183·81-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

\(L(1)\) \(\approx\) \(2.454143937\)
\(L(\frac12)\) \(\approx\) \(2.454143937\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.18T + 3T^{2} \)
11 \( 1 - 0.230T + 11T^{2} \)
13 \( 1 - 2.27T + 13T^{2} \)
17 \( 1 + 6.53T + 17T^{2} \)
19 \( 1 + 0.260T + 19T^{2} \)
23 \( 1 - 8.87T + 23T^{2} \)
29 \( 1 - 5.42T + 29T^{2} \)
31 \( 1 - 4.87T + 31T^{2} \)
37 \( 1 - 1.15T + 37T^{2} \)
41 \( 1 - 4.43T + 41T^{2} \)
43 \( 1 + 4.17T + 43T^{2} \)
47 \( 1 + 0.923T + 47T^{2} \)
53 \( 1 - 2.13T + 53T^{2} \)
59 \( 1 + 5.07T + 59T^{2} \)
61 \( 1 - 8.88T + 61T^{2} \)
67 \( 1 + 8.15T + 67T^{2} \)
71 \( 1 + 9.06T + 71T^{2} \)
73 \( 1 + 6.00T + 73T^{2} \)
79 \( 1 - 0.112T + 79T^{2} \)
83 \( 1 + 8.72T + 83T^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 + 4.29T + 97T^{2} \)
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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

−7.77632408048877517136411995808, −6.93478514173608669051610818423, −6.43298267699836946115476423432, −5.67998255264306391237474520852, −4.76012531650647771736467513749, −4.23329856000612355624231901898, −3.17970564826847972181222906333, −2.77131975209990943993911132757, −1.84682908632839335613690123278, −0.71238887731321545964268806259, 0.71238887731321545964268806259, 1.84682908632839335613690123278, 2.77131975209990943993911132757, 3.17970564826847972181222906333, 4.23329856000612355624231901898, 4.76012531650647771736467513749, 5.67998255264306391237474520852, 6.43298267699836946115476423432, 6.93478514173608669051610818423, 7.77632408048877517136411995808

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