Properties

Label 2-96600-1.1-c1-0-3
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 − 3·11-s − 2·13-s + 19-s + 21-s + 23-s − 27-s − 2·31-s + 3·33-s − 2·37-s + 2·39-s − 5·41-s + 8·43-s − 7·47-s + 49-s + 9·53-s − 57-s − 4·59-s + 2·61-s − 63-s − 4·67-s − 69-s − 8·71-s − 4·73-s + 3·77-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.554·13-s + 0.229·19-s + 0.218·21-s + 0.208·23-s − 0.192·27-s − 0.359·31-s + 0.522·33-s − 0.328·37-s + 0.320·39-s − 0.780·41-s + 1.21·43-s − 1.02·47-s + 1/7·49-s + 1.23·53-s − 0.132·57-s − 0.520·59-s + 0.256·61-s − 0.125·63-s − 0.488·67-s − 0.120·69-s − 0.949·71-s − 0.468·73-s + 0.341·77-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

\(L(1)\) \(\approx\) \(0.6882114072\)
\(L(\frac12)\) \(\approx\) \(0.6882114072\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.67226411106326, −13.13836199739928, −12.91776779943497, −12.27079935325600, −11.87212721378330, −11.38505204077480, −10.77547881955018, −10.30667564468580, −10.03710158606284, −9.324377866813700, −8.926122420572910, −8.212051893051540, −7.691260812409179, −7.167524736211924, −6.791218467921957, −6.026647655575965, −5.626931143255791, −5.072295488757222, −4.599551314188767, −3.919876283097133, −3.218058443139521, −2.665289471431796, −1.990947807016523, −1.176595179005456, −0.2865800599326256, 0.2865800599326256, 1.176595179005456, 1.990947807016523, 2.665289471431796, 3.218058443139521, 3.919876283097133, 4.599551314188767, 5.072295488757222, 5.626931143255791, 6.026647655575965, 6.791218467921957, 7.167524736211924, 7.691260812409179, 8.212051893051540, 8.926122420572910, 9.324377866813700, 10.03710158606284, 10.30667564468580, 10.77547881955018, 11.38505204077480, 11.87212721378330, 12.27079935325600, 12.91776779943497, 13.13836199739928, 13.67226411106326

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