Properties

Label 2-9800-1.1-c1-0-134
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.69·3-s + 4.28·9-s + 0.919·11-s + 5.24·13-s + 4.25·17-s + 1.83·19-s − 8.78·23-s − 3.46·27-s + 6.71·29-s − 4.64·31-s − 2.48·33-s − 1.42·37-s − 14.1·39-s − 7.35·41-s − 9.80·43-s − 6.80·47-s − 11.4·51-s − 11.2·53-s − 4.96·57-s + 11.3·59-s − 9.64·61-s − 2.78·67-s + 23.6·69-s + 11.8·71-s + 5.11·73-s + 0.727·79-s − 3.50·81-s + ⋯
L(s)  = 1  − 1.55·3-s + 1.42·9-s + 0.277·11-s + 1.45·13-s + 1.03·17-s + 0.422·19-s − 1.83·23-s − 0.666·27-s + 1.24·29-s − 0.833·31-s − 0.432·33-s − 0.233·37-s − 2.26·39-s − 1.14·41-s − 1.49·43-s − 0.992·47-s − 1.60·51-s − 1.54·53-s − 0.657·57-s + 1.47·59-s − 1.23·61-s − 0.340·67-s + 2.85·69-s + 1.40·71-s + 0.598·73-s + 0.0818·79-s − 0.389·81-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: \(1\)
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 \( 1 + 2.69T + 3T^{2} \)
11 \( 1 - 0.919T + 11T^{2} \)
13 \( 1 - 5.24T + 13T^{2} \)
17 \( 1 - 4.25T + 17T^{2} \)
19 \( 1 - 1.83T + 19T^{2} \)
23 \( 1 + 8.78T + 23T^{2} \)
29 \( 1 - 6.71T + 29T^{2} \)
31 \( 1 + 4.64T + 31T^{2} \)
37 \( 1 + 1.42T + 37T^{2} \)
41 \( 1 + 7.35T + 41T^{2} \)
43 \( 1 + 9.80T + 43T^{2} \)
47 \( 1 + 6.80T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 9.64T + 61T^{2} \)
67 \( 1 + 2.78T + 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 - 5.11T + 73T^{2} \)
79 \( 1 - 0.727T + 79T^{2} \)
83 \( 1 + 4.37T + 83T^{2} \)
89 \( 1 + 0.963T + 89T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−7.10315262642980619530344187753, −6.31875323483475148755536205834, −6.14936156243396906212926488692, −5.31534944183581175638275025509, −4.81150061698239449107378155803, −3.82132625246977680582624952390, −3.30552136656473881136606739185, −1.77614610865345883368166791147, −1.11024909154849437743405045192, 0, 1.11024909154849437743405045192, 1.77614610865345883368166791147, 3.30552136656473881136606739185, 3.82132625246977680582624952390, 4.81150061698239449107378155803, 5.31534944183581175638275025509, 6.14936156243396906212926488692, 6.31875323483475148755536205834, 7.10315262642980619530344187753

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