Properties

Label 2-3800-1.1-c1-0-46
Degree $2$
Conductor $3800$
Sign $-1$
Analytic cond. $30.3431$
Root an. cond. $5.50846$
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  − 2.73·3-s + 4.46·9-s + 2·11-s + 0.732·13-s − 7.46·17-s + 19-s + 1.46·23-s − 3.99·27-s − 3.46·29-s + 4·31-s − 5.46·33-s − 7.66·37-s − 2·39-s + 0.535·41-s + 2.92·43-s + 2.92·47-s − 7·49-s + 20.3·51-s + 11.6·53-s − 2.73·57-s + 1.46·59-s − 6.53·61-s + 4.19·67-s − 4·69-s + 10.9·71-s + 10.3·73-s − 8.39·79-s + ⋯
L(s)  = 1  − 1.57·3-s + 1.48·9-s + 0.603·11-s + 0.203·13-s − 1.81·17-s + 0.229·19-s + 0.305·23-s − 0.769·27-s − 0.643·29-s + 0.718·31-s − 0.951·33-s − 1.25·37-s − 0.320·39-s + 0.0836·41-s + 0.446·43-s + 0.427·47-s − 49-s + 2.85·51-s + 1.60·53-s − 0.361·57-s + 0.190·59-s − 0.836·61-s + 0.512·67-s − 0.481·69-s + 1.29·71-s + 1.21·73-s − 0.944·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(30.3431\)
Root analytic conductor: \(5.50846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + 2.73T + 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 0.732T + 13T^{2} \)
17 \( 1 + 7.46T + 17T^{2} \)
23 \( 1 - 1.46T + 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 7.66T + 37T^{2} \)
41 \( 1 - 0.535T + 41T^{2} \)
43 \( 1 - 2.92T + 43T^{2} \)
47 \( 1 - 2.92T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 - 1.46T + 59T^{2} \)
61 \( 1 + 6.53T + 61T^{2} \)
67 \( 1 - 4.19T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 - 10.3T + 73T^{2} \)
79 \( 1 + 8.39T + 79T^{2} \)
83 \( 1 - 5.46T + 83T^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
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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

−8.108001808058797947298978454859, −6.90456756639894863252033783107, −6.74075841860756512839471624831, −5.90381953976311534480782780472, −5.18248248265521182432674123906, −4.48327970383893913748327302728, −3.71748707450326020536028965323, −2.30547136571454559142166571311, −1.15754762934150027273408769744, 0, 1.15754762934150027273408769744, 2.30547136571454559142166571311, 3.71748707450326020536028965323, 4.48327970383893913748327302728, 5.18248248265521182432674123906, 5.90381953976311534480782780472, 6.74075841860756512839471624831, 6.90456756639894863252033783107, 8.108001808058797947298978454859

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