Properties

Label 2-96600-1.1-c1-0-33
Degree $2$
Conductor $96600$
Sign $1$
Analytic cond. $771.354$
Root an. cond. $27.7732$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s + 6·11-s + 2·13-s + 4·17-s + 2·19-s − 21-s − 23-s + 27-s − 8·29-s + 6·31-s + 6·33-s − 2·37-s + 2·39-s − 10·41-s + 2·43-s + 2·47-s + 49-s + 4·51-s + 6·53-s + 2·57-s + 4·59-s + 6·61-s − 63-s + 2·67-s − 69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.80·11-s + 0.554·13-s + 0.970·17-s + 0.458·19-s − 0.218·21-s − 0.208·23-s + 0.192·27-s − 1.48·29-s + 1.07·31-s + 1.04·33-s − 0.328·37-s + 0.320·39-s − 1.56·41-s + 0.304·43-s + 0.291·47-s + 1/7·49-s + 0.560·51-s + 0.824·53-s + 0.264·57-s + 0.520·59-s + 0.768·61-s − 0.125·63-s + 0.244·67-s − 0.120·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96600\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(771.354\)
Root analytic conductor: \(27.7732\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 96600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.320896188\)
\(L(\frac12)\) \(\approx\) \(4.320896188\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
23 \( 1 + T \)
good11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.86554885816439, −13.46933736094274, −12.79578176200686, −12.31432008972246, −11.83002641644823, −11.47996488617635, −10.86956831096514, −10.14840538404555, −9.716485332526704, −9.417421409241185, −8.756289257834815, −8.438842411232679, −7.789846594587224, −7.211324106984737, −6.661678972750786, −6.338870200311300, −5.558577673569416, −5.147865142000304, −4.157026764718644, −3.777235325051336, −3.465406483593388, −2.709533389104673, −1.885107153615529, −1.315489702025182, −0.6801064411080611, 0.6801064411080611, 1.315489702025182, 1.885107153615529, 2.709533389104673, 3.465406483593388, 3.777235325051336, 4.157026764718644, 5.147865142000304, 5.558577673569416, 6.338870200311300, 6.661678972750786, 7.211324106984737, 7.789846594587224, 8.438842411232679, 8.756289257834815, 9.417421409241185, 9.716485332526704, 10.14840538404555, 10.86956831096514, 11.47996488617635, 11.83002641644823, 12.31432008972246, 12.79578176200686, 13.46933736094274, 13.86554885816439

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