Properties

Label 2-9800-1.1-c1-0-139
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  − 1.34·3-s − 1.18·9-s + 1.41·11-s + 6.47·13-s − 2.59·17-s − 1.16·19-s − 1.57·23-s + 5.63·27-s + 6.22·29-s − 4.98·31-s − 1.90·33-s − 8.47·37-s − 8.72·39-s − 3.57·41-s − 7.24·43-s − 10.9·47-s + 3.49·51-s + 12.7·53-s + 1.56·57-s + 2.04·59-s + 0.615·61-s + 6.08·67-s + 2.12·69-s − 8.27·71-s − 3.51·73-s + 0.879·79-s − 4.04·81-s + ⋯
L(s)  = 1  − 0.777·3-s − 0.394·9-s + 0.425·11-s + 1.79·13-s − 0.629·17-s − 0.266·19-s − 0.328·23-s + 1.08·27-s + 1.15·29-s − 0.895·31-s − 0.331·33-s − 1.39·37-s − 1.39·39-s − 0.558·41-s − 1.10·43-s − 1.59·47-s + 0.489·51-s + 1.75·53-s + 0.207·57-s + 0.265·59-s + 0.0788·61-s + 0.743·67-s + 0.255·69-s − 0.982·71-s − 0.411·73-s + 0.0989·79-s − 0.449·81-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 \( 1 + 1.34T + 3T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
13 \( 1 - 6.47T + 13T^{2} \)
17 \( 1 + 2.59T + 17T^{2} \)
19 \( 1 + 1.16T + 19T^{2} \)
23 \( 1 + 1.57T + 23T^{2} \)
29 \( 1 - 6.22T + 29T^{2} \)
31 \( 1 + 4.98T + 31T^{2} \)
37 \( 1 + 8.47T + 37T^{2} \)
41 \( 1 + 3.57T + 41T^{2} \)
43 \( 1 + 7.24T + 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 - 12.7T + 53T^{2} \)
59 \( 1 - 2.04T + 59T^{2} \)
61 \( 1 - 0.615T + 61T^{2} \)
67 \( 1 - 6.08T + 67T^{2} \)
71 \( 1 + 8.27T + 71T^{2} \)
73 \( 1 + 3.51T + 73T^{2} \)
79 \( 1 - 0.879T + 79T^{2} \)
83 \( 1 - 3.90T + 83T^{2} \)
89 \( 1 + 5.96T + 89T^{2} \)
97 \( 1 + 3.23T + 97T^{2} \)
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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

−7.02952304452058724584864392355, −6.56838860280185303976787371920, −6.02807234855834406109835748668, −5.37412612683603453426028224073, −4.65354387789043841084178215480, −3.77556959055943832267274248422, −3.20383294196903007490161382417, −2.00220506698059731288563630351, −1.13131983885756149735412009720, 0, 1.13131983885756149735412009720, 2.00220506698059731288563630351, 3.20383294196903007490161382417, 3.77556959055943832267274248422, 4.65354387789043841084178215480, 5.37412612683603453426028224073, 6.02807234855834406109835748668, 6.56838860280185303976787371920, 7.02952304452058724584864392355

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