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Properties

Label 2-96600-1.1-c1-0-12
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 − 8·19-s − 21-s − 23-s − 27-s − 2·29-s − 4·31-s − 4·33-s − 6·37-s + 2·39-s − 6·41-s − 4·43-s + 12·47-s + 49-s − 2·51-s + 10·53-s + 8·57-s + 12·59-s − 2·61-s + 63-s + 4·67-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 − 1.83·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.696·33-s − 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 1.75·47-s + 1/7·49-s − 0.280·51-s + 1.37·53-s + 1.05·57-s + 1.56·59-s − 0.256·61-s + 0.125·63-s + 0.488·67-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

\(L(1)\) \(\approx\) \(1.539746030\)
\(L(\frac12)\) \(\approx\) \(1.539746030\)
\(L(\frac{3}{2})\) 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 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.85385611563555, −13.26905590460719, −12.62810181548566, −12.30894297994236, −11.85263049766252, −11.39302097011552, −10.87547120706431, −10.31847830232686, −10.01498937915867, −9.302923039551537, −8.643965998059308, −8.568458367620197, −7.600683532266744, −7.213567415682710, −6.589993004435005, −6.292008154693770, −5.473152653492972, −5.187543480682967, −4.409374871105098, −3.907084150464433, −3.544513398912630, −2.388838546312551, −1.981219597089130, −1.252527224640618, −0.4228754292837018, 0.4228754292837018, 1.252527224640618, 1.981219597089130, 2.388838546312551, 3.544513398912630, 3.907084150464433, 4.409374871105098, 5.187543480682967, 5.473152653492972, 6.292008154693770, 6.589993004435005, 7.213567415682710, 7.600683532266744, 8.568458367620197, 8.643965998059308, 9.302923039551537, 10.01498937915867, 10.31847830232686, 10.87547120706431, 11.39302097011552, 11.85263049766252, 12.30894297994236, 12.62810181548566, 13.26905590460719, 13.85385611563555

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