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Properties

Label 2-92400-1.1-c1-0-11
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s + 9-s − 11-s − 4·13-s + 4·17-s + 2·19-s − 21-s − 6·23-s − 27-s − 2·29-s + 2·31-s + 33-s − 2·37-s + 4·39-s − 12·41-s + 5·43-s − 3·47-s + 49-s − 4·51-s − 7·53-s − 2·57-s − 6·59-s + 14·61-s + 63-s + 6·69-s − 12·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.970·17-s + 0.458·19-s − 0.218·21-s − 1.25·23-s − 0.192·27-s − 0.371·29-s + 0.359·31-s + 0.174·33-s − 0.328·37-s + 0.640·39-s − 1.87·41-s + 0.762·43-s − 0.437·47-s + 1/7·49-s − 0.560·51-s − 0.961·53-s − 0.264·57-s − 0.781·59-s + 1.79·61-s + 0.125·63-s + 0.722·69-s − 1.42·71-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: \(0\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9397312377\)
\(L(\frac12)\) \(\approx\) \(0.9397312377\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 7 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−13.87373334467225, −13.38752569365181, −12.66197786671861, −12.29904565563633, −11.93297363858777, −11.44488183083453, −10.95035216394748, −10.21768660173123, −9.979131404990466, −9.592947999209770, −8.833709747390468, −8.121577611082475, −7.845652253157691, −7.259012145376469, −6.765601192424893, −6.112306710329881, −5.480767520800588, −5.179025720043344, −4.607668381343193, −3.951803983275381, −3.309489828138253, −2.618659081727773, −1.891939059819723, −1.296913182299253, −0.3246176348302833, 0.3246176348302833, 1.296913182299253, 1.891939059819723, 2.618659081727773, 3.309489828138253, 3.951803983275381, 4.607668381343193, 5.179025720043344, 5.480767520800588, 6.112306710329881, 6.765601192424893, 7.259012145376469, 7.845652253157691, 8.121577611082475, 8.833709747390468, 9.592947999209770, 9.979131404990466, 10.21768660173123, 10.95035216394748, 11.44488183083453, 11.93297363858777, 12.29904565563633, 12.66197786671861, 13.38752569365181, 13.87373334467225

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