Properties

Label 2-92400-1.1-c1-0-125
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 + 21-s + 6·23-s − 27-s − 2·31-s + 33-s + 8·37-s − 4·39-s + 2·41-s − 4·43-s − 8·47-s + 49-s + 2·51-s − 6·53-s + 2·61-s − 63-s − 8·67-s − 6·69-s + 8·71-s + 4·73-s + 77-s − 10·79-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 + 0.218·21-s + 1.25·23-s − 0.192·27-s − 0.359·31-s + 0.174·33-s + 1.31·37-s − 0.640·39-s + 0.312·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 0.256·61-s − 0.125·63-s − 0.977·67-s − 0.722·69-s + 0.949·71-s + 0.468·73-s + 0.113·77-s − 1.12·79-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: $\chi_{92400} (1, \cdot )$
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 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−13.99965572017161, −13.40116720271445, −13.03774539435623, −12.79844924562044, −12.12247366507886, −11.39080680238607, −11.23479764059552, −10.74321768391486, −10.19800441198268, −9.602031977391279, −9.164564377189097, −8.601670611982154, −8.077787908311640, −7.459936365404792, −6.910429848566441, −6.318960694988572, −6.072645221924384, −5.318728592737126, −4.833929684077254, −4.260918605946001, −3.553682612841675, −3.068944959023271, −2.318787378829486, −1.487917841756550, −0.8649593008575835, 0, 0.8649593008575835, 1.487917841756550, 2.318787378829486, 3.068944959023271, 3.553682612841675, 4.260918605946001, 4.833929684077254, 5.318728592737126, 6.072645221924384, 6.318960694988572, 6.910429848566441, 7.459936365404792, 8.077787908311640, 8.601670611982154, 9.164564377189097, 9.602031977391279, 10.19800441198268, 10.74321768391486, 11.23479764059552, 11.39080680238607, 12.12247366507886, 12.79844924562044, 13.03774539435623, 13.40116720271445, 13.99965572017161

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