Properties

Label 2-92400-1.1-c1-0-104
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s + 9-s − 11-s − 6·13-s + 6·17-s + 4·19-s + 21-s − 4·23-s − 27-s + 6·29-s + 4·31-s + 33-s − 6·37-s + 6·39-s − 2·41-s − 4·43-s + 49-s − 6·51-s − 14·53-s − 4·57-s − 6·61-s − 63-s − 8·67-s + 4·69-s + 8·71-s + 14·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 1.45·17-s + 0.917·19-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 − 0.986·37-s + 0.960·39-s − 0.312·41-s − 0.609·43-s + 1/7·49-s − 0.840·51-s − 1.92·53-s − 0.529·57-s − 0.768·61-s − 0.125·63-s − 0.977·67-s + 0.481·69-s + 0.949·71-s + 1.63·73-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: $\chi_{92400} (1, \cdot )$
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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.18363493169225, −13.70901723935705, −12.95474554196932, −12.38901984054836, −12.13562608644870, −11.87770884711270, −11.17394450288697, −10.47574888583362, −10.05651466581480, −9.781989583500058, −9.364302503226314, −8.473088389726501, −7.909800359346989, −7.559452983561382, −6.995431777764022, −6.426777714869823, −5.897479883942979, −5.216651358519070, −4.941442646081245, −4.373056138587110, −3.378900554593206, −3.127201099913036, −2.327092216709038, −1.575441919351989, −0.7578186247899586, 0, 0.7578186247899586, 1.575441919351989, 2.327092216709038, 3.127201099913036, 3.378900554593206, 4.373056138587110, 4.941442646081245, 5.216651358519070, 5.897479883942979, 6.426777714869823, 6.995431777764022, 7.559452983561382, 7.909800359346989, 8.473088389726501, 9.364302503226314, 9.781989583500058, 10.05651466581480, 10.47574888583362, 11.17394450288697, 11.87770884711270, 12.13562608644870, 12.38901984054836, 12.95474554196932, 13.70901723935705, 14.18363493169225

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