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Properties

Label 2-92400-1.1-c1-0-136
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 + 2·13-s − 6·17-s + 4·19-s − 21-s − 6·23-s + 27-s − 8·29-s + 4·31-s − 33-s + 8·37-s + 2·39-s + 4·41-s − 5·43-s − 7·47-s + 49-s − 6·51-s + 53-s + 4·57-s + 8·59-s − 4·61-s − 63-s + 6·67-s − 6·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 1.45·17-s + 0.917·19-s − 0.218·21-s − 1.25·23-s + 0.192·27-s − 1.48·29-s + 0.718·31-s − 0.174·33-s + 1.31·37-s + 0.320·39-s + 0.624·41-s − 0.762·43-s − 1.02·47-s + 1/7·49-s − 0.840·51-s + 0.137·53-s + 0.529·57-s + 1.04·59-s − 0.512·61-s − 0.125·63-s + 0.733·67-s − 0.722·69-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: Trivial
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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
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

−14.06399303529723, −13.48356030042065, −13.20826164980687, −12.81168404734360, −12.16524175796351, −11.47798815382803, −11.23833627170210, −10.64319037299116, −9.948697410510926, −9.556696310937606, −9.251438873322557, −8.452787425061787, −8.137377314579365, −7.665568346673366, −6.874804634056672, −6.632852723370749, −5.834645813358585, −5.484554024143451, −4.576369457733629, −4.169887123108005, −3.567088907897824, −2.988201876227601, −2.286846254691165, −1.827245232541386, −0.8864916031610456, 0, 0.8864916031610456, 1.827245232541386, 2.286846254691165, 2.988201876227601, 3.567088907897824, 4.169887123108005, 4.576369457733629, 5.484554024143451, 5.834645813358585, 6.632852723370749, 6.874804634056672, 7.665568346673366, 8.137377314579365, 8.452787425061787, 9.251438873322557, 9.556696310937606, 9.948697410510926, 10.64319037299116, 11.23833627170210, 11.47798815382803, 12.16524175796351, 12.81168404734360, 13.20826164980687, 13.48356030042065, 14.06399303529723

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