Properties

Label 2-62400-1.1-c1-0-41
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

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·7-s + 9-s + 13-s − 3·19-s + 4·21-s + 4·23-s − 27-s + 29-s + 8·31-s + 37-s − 39-s + 41-s + 6·43-s + 11·47-s + 9·49-s + 3·53-s + 3·57-s + 10·59-s − 4·61-s − 4·63-s + 13·67-s − 4·69-s − 9·71-s + 3·79-s + 81-s + 2·83-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.688·19-s + 0.872·21-s + 0.834·23-s − 0.192·27-s + 0.185·29-s + 1.43·31-s + 0.164·37-s − 0.160·39-s + 0.156·41-s + 0.914·43-s + 1.60·47-s + 9/7·49-s + 0.412·53-s + 0.397·57-s + 1.30·59-s − 0.512·61-s − 0.503·63-s + 1.58·67-s − 0.481·69-s − 1.06·71-s + 0.337·79-s + 1/9·81-s + 0.219·83-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

\(L(1)\) \(\approx\) \(1.583851287\)
\(L(\frac12)\) \(\approx\) \(1.583851287\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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

−14.11880350659277, −13.70027401164463, −13.12226265400965, −12.74619998672457, −12.37676210700422, −11.75408725546210, −11.26686384065056, −10.58521149271302, −10.25729301389700, −9.787661350535025, −9.075939452549188, −8.830933649768658, −8.023952418175406, −7.369048173558859, −6.781494577737395, −6.428327351766042, −5.918301117141105, −5.386335018593044, −4.594444861096301, −4.062241610793653, −3.448949929032450, −2.750893911298673, −2.224200972973777, −1.016577134523630, −0.5369697588131837, 0.5369697588131837, 1.016577134523630, 2.224200972973777, 2.750893911298673, 3.448949929032450, 4.062241610793653, 4.594444861096301, 5.386335018593044, 5.918301117141105, 6.428327351766042, 6.781494577737395, 7.369048173558859, 8.023952418175406, 8.830933649768658, 9.075939452549188, 9.787661350535025, 10.25729301389700, 10.58521149271302, 11.26686384065056, 11.75408725546210, 12.37676210700422, 12.74619998672457, 13.12226265400965, 13.70027401164463, 14.11880350659277

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