Properties

Label 2-20e2-1.1-c9-0-13
Degree $2$
Conductor $400$
Sign $1$
Analytic cond. $206.014$
Root an. cond. $14.3531$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 12.2·3-s + 9.62e3·7-s − 1.95e4·9-s − 5.56e4·11-s − 1.69e5·13-s − 2.07e5·17-s − 8.02e5·19-s + 1.17e5·21-s − 1.24e6·23-s − 4.78e5·27-s − 4.28e6·29-s + 3.58e6·31-s − 6.79e5·33-s + 2.89e6·37-s − 2.07e6·39-s + 2.51e7·41-s − 2.00e7·43-s + 3.73e7·47-s + 5.22e7·49-s − 2.53e6·51-s + 2.55e7·53-s − 9.79e6·57-s + 9.96e7·59-s + 2.00e8·61-s − 1.87e8·63-s − 8.09e7·67-s − 1.51e7·69-s + ⋯
L(s)  = 1  + 0.0870·3-s + 1.51·7-s − 0.992·9-s − 1.14·11-s − 1.64·13-s − 0.602·17-s − 1.41·19-s + 0.131·21-s − 0.925·23-s − 0.173·27-s − 1.12·29-s + 0.697·31-s − 0.0997·33-s + 0.254·37-s − 0.143·39-s + 1.39·41-s − 0.893·43-s + 1.11·47-s + 1.29·49-s − 0.0524·51-s + 0.444·53-s − 0.122·57-s + 1.07·59-s + 1.85·61-s − 1.50·63-s − 0.491·67-s − 0.0805·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(206.014\)
Root analytic conductor: \(14.3531\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(1.174714189\)
\(L(\frac12)\) \(\approx\) \(1.174714189\)
\(L(\frac{11}{2})\) 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 \( 1 \)
good3 \( 1 - 12.2T + 1.96e4T^{2} \)
7 \( 1 - 9.62e3T + 4.03e7T^{2} \)
11 \( 1 + 5.56e4T + 2.35e9T^{2} \)
13 \( 1 + 1.69e5T + 1.06e10T^{2} \)
17 \( 1 + 2.07e5T + 1.18e11T^{2} \)
19 \( 1 + 8.02e5T + 3.22e11T^{2} \)
23 \( 1 + 1.24e6T + 1.80e12T^{2} \)
29 \( 1 + 4.28e6T + 1.45e13T^{2} \)
31 \( 1 - 3.58e6T + 2.64e13T^{2} \)
37 \( 1 - 2.89e6T + 1.29e14T^{2} \)
41 \( 1 - 2.51e7T + 3.27e14T^{2} \)
43 \( 1 + 2.00e7T + 5.02e14T^{2} \)
47 \( 1 - 3.73e7T + 1.11e15T^{2} \)
53 \( 1 - 2.55e7T + 3.29e15T^{2} \)
59 \( 1 - 9.96e7T + 8.66e15T^{2} \)
61 \( 1 - 2.00e8T + 1.16e16T^{2} \)
67 \( 1 + 8.09e7T + 2.72e16T^{2} \)
71 \( 1 - 4.31e7T + 4.58e16T^{2} \)
73 \( 1 - 3.40e8T + 5.88e16T^{2} \)
79 \( 1 + 2.81e8T + 1.19e17T^{2} \)
83 \( 1 + 6.01e8T + 1.86e17T^{2} \)
89 \( 1 - 5.39e8T + 3.50e17T^{2} \)
97 \( 1 + 4.23e8T + 7.60e17T^{2} \)
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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

−9.824699251864237703822195011254, −8.575573685387857101593383689217, −8.057751739306022178482963632140, −7.20878395773451171769088877321, −5.78140983981250689512392324572, −5.01962688825234605250198325793, −4.18656834028466691642015723888, −2.43541760848382070913734855595, −2.16195920886305474303606885900, −0.41762187556954241902528467500, 0.41762187556954241902528467500, 2.16195920886305474303606885900, 2.43541760848382070913734855595, 4.18656834028466691642015723888, 5.01962688825234605250198325793, 5.78140983981250689512392324572, 7.20878395773451171769088877321, 8.057751739306022178482963632140, 8.575573685387857101593383689217, 9.824699251864237703822195011254

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