Properties

Label 2-62400-1.1-c1-0-68
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 − 7-s + 9-s + 5·11-s + 13-s + 5·17-s − 21-s + 27-s + 7·29-s − 9·31-s + 5·33-s + 8·37-s + 39-s − 2·41-s − 8·43-s − 9·47-s − 6·49-s + 5·51-s − 11·53-s − 59-s + 7·61-s − 63-s + 15·67-s − 8·71-s + 4·73-s − 5·77-s − 4·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s + 1.21·17-s − 0.218·21-s + 0.192·27-s + 1.29·29-s − 1.61·31-s + 0.870·33-s + 1.31·37-s + 0.160·39-s − 0.312·41-s − 1.21·43-s − 1.31·47-s − 6/7·49-s + 0.700·51-s − 1.51·53-s − 0.130·59-s + 0.896·61-s − 0.125·63-s + 1.83·67-s − 0.949·71-s + 0.468·73-s − 0.569·77-s − 0.450·79-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

\(L(1)\) \(\approx\) \(3.722655230\)
\(L(\frac12)\) \(\approx\) \(3.722655230\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−14.25697502295603, −13.99105787772092, −13.10708630291239, −12.92074439445039, −12.27746793095830, −11.71462397067128, −11.37932979901258, −10.66920399489652, −10.00493905663836, −9.587249112360242, −9.271507447753028, −8.584487363135226, −8.089693750145444, −7.639990144847505, −6.800058652759066, −6.529863678909941, −5.979011412395604, −5.157251865669576, −4.625779575545631, −3.810703347577152, −3.446005802996607, −2.971826800282152, −1.945736729837822, −1.425442858788866, −0.6596305344431028, 0.6596305344431028, 1.425442858788866, 1.945736729837822, 2.971826800282152, 3.446005802996607, 3.810703347577152, 4.625779575545631, 5.157251865669576, 5.979011412395604, 6.529863678909941, 6.800058652759066, 7.639990144847505, 8.089693750145444, 8.584487363135226, 9.271507447753028, 9.587249112360242, 10.00493905663836, 10.66920399489652, 11.37932979901258, 11.71462397067128, 12.27746793095830, 12.92074439445039, 13.10708630291239, 13.99105787772092, 14.25697502295603

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