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Properties

Label 2-96600-1.1-c1-0-58
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s − 4·11-s + 2·13-s + 5·19-s − 21-s + 23-s + 27-s + 6·29-s − 2·31-s − 4·33-s + 2·37-s + 2·39-s + 10·41-s + 2·43-s − 3·47-s + 49-s − 2·53-s + 5·57-s − 3·59-s + 61-s − 63-s − 16·67-s + 69-s + 6·71-s − 8·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 1.14·19-s − 0.218·21-s + 0.208·23-s + 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.696·33-s + 0.328·37-s + 0.320·39-s + 1.56·41-s + 0.304·43-s − 0.437·47-s + 1/7·49-s − 0.274·53-s + 0.662·57-s − 0.390·59-s + 0.128·61-s − 0.125·63-s − 1.95·67-s + 0.120·69-s + 0.712·71-s − 0.936·73-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: \(1\)
Selberg data: \((2,\ 96600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−13.87402316933241, −13.61129396934360, −13.13396039528746, −12.59144860507426, −12.29746914982204, −11.45686661026189, −11.14434357723280, −10.41167204317685, −10.17499207272288, −9.507289209071005, −9.110177618590637, −8.549768786826446, −7.957041538774305, −7.622953050980275, −7.073221345448220, −6.478747480457736, −5.734576683061237, −5.478833297573338, −4.613330995622723, −4.230296793228831, −3.404138654800476, −2.846432777114160, −2.617954637131893, −1.587045080564854, −0.9704530074266643, 0, 0.9704530074266643, 1.587045080564854, 2.617954637131893, 2.846432777114160, 3.404138654800476, 4.230296793228831, 4.613330995622723, 5.478833297573338, 5.734576683061237, 6.478747480457736, 7.073221345448220, 7.622953050980275, 7.957041538774305, 8.549768786826446, 9.110177618590637, 9.507289209071005, 10.17499207272288, 10.41167204317685, 11.14434357723280, 11.45686661026189, 12.29746914982204, 12.59144860507426, 13.13396039528746, 13.61129396934360, 13.87402316933241

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