Properties

Label 2-96600-1.1-c1-0-8
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 − 4·11-s + 2·13-s − 2·17-s − 4·19-s + 21-s + 23-s − 27-s − 2·29-s + 8·31-s + 4·33-s − 6·37-s − 2·39-s − 6·41-s + 4·43-s − 8·47-s + 49-s + 2·51-s + 10·53-s + 4·57-s − 12·59-s + 14·61-s − 63-s + 12·67-s − 69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.218·21-s + 0.208·23-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s − 0.986·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.280·51-s + 1.37·53-s + 0.529·57-s − 1.56·59-s + 1.79·61-s − 0.125·63-s + 1.46·67-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: $\chi_{96600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 96600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9892302698\)
\(L(\frac12)\) \(\approx\) \(0.9892302698\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−13.75532417956138, −13.20501705725840, −12.83599432222031, −12.38530844206382, −11.86147456150915, −11.22370351797303, −10.79963868919239, −10.50659582277175, −9.848835085493724, −9.509224578440437, −8.666854803743798, −8.290498433234922, −7.880884070410715, −7.070294860987797, −6.531868967510770, −6.373153699381710, −5.418519702133134, −5.200157323074290, −4.567066522484234, −3.835220026294884, −3.392010555293852, −2.456352199385230, −2.141013546394700, −1.111475570777908, −0.3556359444731770, 0.3556359444731770, 1.111475570777908, 2.141013546394700, 2.456352199385230, 3.392010555293852, 3.835220026294884, 4.567066522484234, 5.200157323074290, 5.418519702133134, 6.373153699381710, 6.531868967510770, 7.070294860987797, 7.880884070410715, 8.290498433234922, 8.666854803743798, 9.509224578440437, 9.848835085493724, 10.50659582277175, 10.79963868919239, 11.22370351797303, 11.86147456150915, 12.38530844206382, 12.83599432222031, 13.20501705725840, 13.75532417956138

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