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Properties

Label 2-92400-1.1-c1-0-137
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 + 2·17-s − 21-s − 4·23-s − 27-s + 6·29-s − 4·31-s − 33-s + 10·37-s + 2·39-s − 2·41-s − 4·43-s + 8·47-s + 49-s − 2·51-s − 2·53-s − 4·59-s − 10·61-s + 63-s + 4·67-s + 4·69-s − 8·71-s + 10·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.485·17-s − 0.218·21-s − 0.834·23-s − 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.174·33-s + 1.64·37-s + 0.320·39-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.274·53-s − 0.520·59-s − 1.28·61-s + 0.125·63-s + 0.488·67-s + 0.481·69-s − 0.949·71-s + 1.17·73-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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.10218357727759, −13.65329309280433, −13.05661267593188, −12.34601299252890, −12.23084268156288, −11.65851946238850, −11.18648913019899, −10.57938268217366, −10.27549052557393, −9.500514184672283, −9.359042597580770, −8.477847042848962, −8.012585102311919, −7.522546840902712, −7.030137476118919, −6.213333129037034, −6.104354577350854, −5.221577986854647, −4.899035653445341, −4.216065620460344, −3.752499272643267, −2.902718358582428, −2.301437589954018, −1.530831921936055, −0.8889676159827241, 0, 0.8889676159827241, 1.530831921936055, 2.301437589954018, 2.902718358582428, 3.752499272643267, 4.216065620460344, 4.899035653445341, 5.221577986854647, 6.104354577350854, 6.213333129037034, 7.030137476118919, 7.522546840902712, 8.012585102311919, 8.477847042848962, 9.359042597580770, 9.500514184672283, 10.27549052557393, 10.57938268217366, 11.18648913019899, 11.65851946238850, 12.23084268156288, 12.34601299252890, 13.05661267593188, 13.65329309280433, 14.10218357727759

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