Properties

Label 2-62400-1.1-c1-0-72
Degree $2$
Conductor $62400$
Sign $1$
Analytic cond. $498.266$
Root an. cond. $22.3218$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·7-s + 9-s + 4·11-s + 13-s + 4·17-s + 7·19-s + 4·21-s + 4·23-s − 27-s − 5·29-s − 4·31-s − 4·33-s + 9·37-s − 39-s − 5·41-s + 10·43-s + 3·47-s + 9·49-s − 4·51-s + 9·53-s − 7·57-s − 6·59-s − 4·61-s − 4·63-s + 7·67-s − 4·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.970·17-s + 1.60·19-s + 0.872·21-s + 0.834·23-s − 0.192·27-s − 0.928·29-s − 0.718·31-s − 0.696·33-s + 1.47·37-s − 0.160·39-s − 0.780·41-s + 1.52·43-s + 0.437·47-s + 9/7·49-s − 0.560·51-s + 1.23·53-s − 0.927·57-s − 0.781·59-s − 0.512·61-s − 0.503·63-s + 0.855·67-s − 0.481·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(62400\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(498.266\)
Root analytic conductor: \(22.3218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{62400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 62400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.296804348\)
\(L(\frac12)\) \(\approx\) \(2.296804348\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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

−14.28288703820304, −13.72513080127682, −13.10769017566679, −12.78568545193506, −12.25484711053318, −11.70909621897534, −11.38521874847299, −10.72669752894432, −10.08087674411045, −9.663297083632838, −9.193727736851825, −8.947691579885373, −7.890756857110215, −7.243692044377423, −7.085719905117632, −6.255130848106816, −5.879688483036331, −5.480343749031375, −4.650488516678868, −3.904031168348960, −3.412391930111490, −3.028058262808065, −2.001168661548800, −1.014013621698754, −0.6771369952895746, 0.6771369952895746, 1.014013621698754, 2.001168661548800, 3.028058262808065, 3.412391930111490, 3.904031168348960, 4.650488516678868, 5.480343749031375, 5.879688483036331, 6.255130848106816, 7.085719905117632, 7.243692044377423, 7.890756857110215, 8.947691579885373, 9.193727736851825, 9.663297083632838, 10.08087674411045, 10.72669752894432, 11.38521874847299, 11.70909621897534, 12.25484711053318, 12.78568545193506, 13.10769017566679, 13.72513080127682, 14.28288703820304

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