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Properties

Label 2-96600-1.1-c1-0-14
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 − 2·17-s + 2·19-s + 21-s + 23-s + 27-s + 8·31-s − 4·33-s − 6·37-s − 8·41-s − 8·43-s + 49-s − 2·51-s − 6·53-s + 2·57-s − 2·59-s − 2·61-s + 63-s − 8·67-s + 69-s + 10·71-s + 6·73-s − 4·77-s + 14·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.485·17-s + 0.458·19-s + 0.218·21-s + 0.208·23-s + 0.192·27-s + 1.43·31-s − 0.696·33-s − 0.986·37-s − 1.24·41-s − 1.21·43-s + 1/7·49-s − 0.280·51-s − 0.824·53-s + 0.264·57-s − 0.260·59-s − 0.256·61-s + 0.125·63-s − 0.977·67-s + 0.120·69-s + 1.18·71-s + 0.702·73-s − 0.455·77-s + 1.57·79-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

\(L(1)\) \(\approx\) \(2.287637261\)
\(L(\frac12)\) \(\approx\) \(2.287637261\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.71720642303708, −13.40704001103955, −12.99073176249301, −12.23818503279682, −11.95990485567529, −11.31676233836937, −10.73856993922861, −10.31728992230546, −9.915956502532579, −9.259587650471122, −8.765252182822392, −8.199040764234933, −7.891220986238478, −7.394402077061531, −6.619552078920825, −6.372669376968232, −5.376493442958652, −4.979957116872899, −4.645557930377839, −3.716683671180573, −3.251094494886688, −2.634207011089590, −2.048819754736518, −1.397867724418591, −0.4507023403918553, 0.4507023403918553, 1.397867724418591, 2.048819754736518, 2.634207011089590, 3.251094494886688, 3.716683671180573, 4.645557930377839, 4.979957116872899, 5.376493442958652, 6.372669376968232, 6.619552078920825, 7.394402077061531, 7.891220986238478, 8.199040764234933, 8.765252182822392, 9.259587650471122, 9.915956502532579, 10.31728992230546, 10.73856993922861, 11.31676233836937, 11.95990485567529, 12.23818503279682, 12.99073176249301, 13.40704001103955, 13.71720642303708

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