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Properties

Label 2-92400-1.1-c1-0-91
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 + 6·19-s + 21-s − 4·23-s − 27-s − 4·29-s − 33-s − 6·37-s + 4·39-s − 8·43-s − 4·47-s + 49-s + 2·51-s + 6·53-s − 6·57-s − 4·59-s + 6·61-s − 63-s − 4·67-s + 4·69-s − 77-s + 6·79-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 + 1.37·19-s + 0.218·21-s − 0.834·23-s − 0.192·27-s − 0.742·29-s − 0.174·33-s − 0.986·37-s + 0.640·39-s − 1.21·43-s − 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.794·57-s − 0.520·59-s + 0.768·61-s − 0.125·63-s − 0.488·67-s + 0.481·69-s − 0.113·77-s + 0.675·79-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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
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 \( 1 + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−13.90726011524836, −13.65545213118862, −13.09517025808205, −12.47412825568492, −12.06386170762662, −11.72600960275681, −11.24596401347184, −10.57406936876601, −10.08757944811278, −9.673598171590706, −9.269470871260992, −8.631630407828473, −7.938594398169231, −7.450575775700245, −6.938095649819997, −6.536828472472247, −5.838122107210834, −5.324692055047702, −4.899038000864746, −4.246297177178955, −3.558131226781829, −3.089745959681960, −2.184759972825479, −1.696050698437317, −0.7198844058765291, 0, 0.7198844058765291, 1.696050698437317, 2.184759972825479, 3.089745959681960, 3.558131226781829, 4.246297177178955, 4.899038000864746, 5.324692055047702, 5.838122107210834, 6.536828472472247, 6.938095649819997, 7.450575775700245, 7.938594398169231, 8.631630407828473, 9.269470871260992, 9.673598171590706, 10.08757944811278, 10.57406936876601, 11.24596401347184, 11.72600960275681, 12.06386170762662, 12.47412825568492, 13.09517025808205, 13.65545213118862, 13.90726011524836

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