Properties

Label 2-92400-1.1-c1-0-118
Degree $2$
Conductor $92400$
Sign $-1$
Analytic cond. $737.817$
Root an. cond. $27.1628$
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 − 11-s + 4·13-s − 2·17-s − 4·19-s − 21-s + 2·23-s − 27-s + 2·31-s + 33-s − 8·37-s − 4·39-s − 6·41-s − 4·43-s + 49-s + 2·51-s − 6·53-s + 4·57-s − 4·59-s + 6·61-s + 63-s − 4·67-s − 2·69-s − 4·73-s − 77-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.485·17-s − 0.917·19-s − 0.218·21-s + 0.417·23-s − 0.192·27-s + 0.359·31-s + 0.174·33-s − 1.31·37-s − 0.640·39-s − 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 0.529·57-s − 0.520·59-s + 0.768·61-s + 0.125·63-s − 0.488·67-s − 0.240·69-s − 0.468·73-s − 0.113·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(737.817\)
Root analytic conductor: \(27.1628\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{92400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−13.93759491437337, −13.56168089230718, −13.11807978028288, −12.61855535841713, −12.05628407114240, −11.63196629956976, −10.95519083820875, −10.81784878093755, −10.28769601063755, −9.680924125831706, −9.023419609865983, −8.475152130425521, −8.243059212460325, −7.481595619038051, −6.817614662928031, −6.514826328282380, −5.901030369226851, −5.356306644523892, −4.753992578573046, −4.350685543178801, −3.567672197931290, −3.114125428291560, −2.091548043039168, −1.684968961053635, −0.8348931427246624, 0, 0.8348931427246624, 1.684968961053635, 2.091548043039168, 3.114125428291560, 3.567672197931290, 4.350685543178801, 4.753992578573046, 5.356306644523892, 5.901030369226851, 6.514826328282380, 6.817614662928031, 7.481595619038051, 8.243059212460325, 8.475152130425521, 9.023419609865983, 9.680924125831706, 10.28769601063755, 10.81784878093755, 10.95519083820875, 11.63196629956976, 12.05628407114240, 12.61855535841713, 13.11807978028288, 13.56168089230718, 13.93759491437337

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