Properties

Label 2-92400-1.1-c1-0-123
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 − 4·13-s + 2·17-s − 4·19-s − 21-s − 2·23-s + 27-s + 2·31-s − 33-s + 8·37-s − 4·39-s − 6·41-s + 4·43-s + 49-s + 2·51-s + 6·53-s − 4·57-s − 4·59-s + 6·61-s − 63-s + 4·67-s − 2·69-s + 4·73-s + 77-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 − 0.917·19-s − 0.218·21-s − 0.417·23-s + 0.192·27-s + 0.359·31-s − 0.174·33-s + 1.31·37-s − 0.640·39-s − 0.937·41-s + 0.609·43-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s + 0.768·61-s − 0.125·63-s + 0.488·67-s − 0.240·69-s + 0.468·73-s + 0.113·77-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 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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

−14.14592711977433, −13.55439549777100, −13.13557698910653, −12.61915913521442, −12.24401940233370, −11.72318824992723, −11.11917002450771, −10.34510696604198, −10.23551072428970, −9.518764253324469, −9.245644232262363, −8.529145318020217, −7.983686371716826, −7.695610332449621, −6.992834865711780, −6.541742958426957, −5.955046601415127, −5.249767270874972, −4.788355607070179, −4.048898342621404, −3.681517383958151, −2.713336054657318, −2.542636027819750, −1.787129498595662, −0.8560389112249946, 0, 0.8560389112249946, 1.787129498595662, 2.542636027819750, 2.713336054657318, 3.681517383958151, 4.048898342621404, 4.788355607070179, 5.249767270874972, 5.955046601415127, 6.541742958426957, 6.992834865711780, 7.695610332449621, 7.983686371716826, 8.529145318020217, 9.245644232262363, 9.518764253324469, 10.23551072428970, 10.34510696604198, 11.11917002450771, 11.72318824992723, 12.24401940233370, 12.61915913521442, 13.13557698910653, 13.55439549777100, 14.14592711977433

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