Properties

Label 2-62400-1.1-c1-0-16
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 − 2·11-s − 13-s + 6·17-s + 4·19-s + 4·21-s + 4·23-s − 27-s + 6·29-s − 8·31-s + 2·33-s − 10·37-s + 39-s − 4·41-s + 4·43-s − 6·47-s + 9·49-s − 6·51-s + 6·53-s − 4·57-s − 6·59-s + 6·61-s − 4·63-s − 4·69-s − 10·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 1.45·17-s + 0.917·19-s + 0.872·21-s + 0.834·23-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.348·33-s − 1.64·37-s + 0.160·39-s − 0.624·41-s + 0.609·43-s − 0.875·47-s + 9/7·49-s − 0.840·51-s + 0.824·53-s − 0.529·57-s − 0.781·59-s + 0.768·61-s − 0.503·63-s − 0.481·69-s − 1.18·71-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

\(L(1)\) \(\approx\) \(1.090373568\)
\(L(\frac12)\) \(\approx\) \(1.090373568\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
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

−14.26468204233815, −13.66373758327652, −13.13802603262385, −12.73954330299377, −12.26149328605295, −11.88648219073307, −11.29524253247485, −10.42071059846454, −10.28899196170075, −9.855144169349837, −9.113202757671358, −8.858159182978915, −7.814167377930731, −7.495084445358444, −6.880783817632744, −6.450195386511025, −5.757326302531472, −5.269365463859958, −4.929560744361054, −3.894567818745101, −3.223652915877501, −3.098682981813309, −2.070921376734914, −1.132392787668896, −0.4103653797572404, 0.4103653797572404, 1.132392787668896, 2.070921376734914, 3.098682981813309, 3.223652915877501, 3.894567818745101, 4.929560744361054, 5.269365463859958, 5.757326302531472, 6.450195386511025, 6.880783817632744, 7.495084445358444, 7.814167377930731, 8.858159182978915, 9.113202757671358, 9.855144169349837, 10.28899196170075, 10.42071059846454, 11.29524253247485, 11.88648219073307, 12.26149328605295, 12.73954330299377, 13.13802603262385, 13.66373758327652, 14.26468204233815

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