Properties

Label 2-62400-1.1-c1-0-9
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 − 6·11-s + 13-s + 4·17-s + 2·19-s + 4·21-s − 6·23-s − 27-s + 10·29-s − 4·31-s + 6·33-s − 6·37-s − 39-s + 10·41-s + 8·47-s + 9·49-s − 4·51-s − 6·53-s − 2·57-s − 6·59-s + 6·61-s − 4·63-s + 12·67-s + 6·69-s − 2·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.80·11-s + 0.277·13-s + 0.970·17-s + 0.458·19-s + 0.872·21-s − 1.25·23-s − 0.192·27-s + 1.85·29-s − 0.718·31-s + 1.04·33-s − 0.986·37-s − 0.160·39-s + 1.56·41-s + 1.16·47-s + 9/7·49-s − 0.560·51-s − 0.824·53-s − 0.264·57-s − 0.781·59-s + 0.768·61-s − 0.503·63-s + 1.46·67-s + 0.722·69-s − 0.234·73-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

\(L(1)\) \(\approx\) \(0.7840189550\)
\(L(\frac12)\) \(\approx\) \(0.7840189550\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−14.09363921654168, −13.68776899820697, −13.24846572573878, −12.54892410329767, −12.38440292523640, −12.00595932137533, −11.01859460714406, −10.69608537772919, −10.16064737536441, −9.828435813859468, −9.350089942997679, −8.542465098584158, −7.939770275789257, −7.545117791420853, −6.897500772457154, −6.287160384578027, −5.841472856013184, −5.361183320148293, −4.811866873876481, −3.933225902550163, −3.417499542900161, −2.761567122136218, −2.264132629405907, −1.086415116644853, −0.3504523398205744, 0.3504523398205744, 1.086415116644853, 2.264132629405907, 2.761567122136218, 3.417499542900161, 3.933225902550163, 4.811866873876481, 5.361183320148293, 5.841472856013184, 6.287160384578027, 6.897500772457154, 7.545117791420853, 7.939770275789257, 8.542465098584158, 9.350089942997679, 9.828435813859468, 10.16064737536441, 10.69608537772919, 11.01859460714406, 12.00595932137533, 12.38440292523640, 12.54892410329767, 13.24846572573878, 13.68776899820697, 14.09363921654168

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