Properties

Label 2-51600-1.1-c1-0-17
Degree $2$
Conductor $51600$
Sign $1$
Analytic cond. $412.028$
Root an. cond. $20.2984$
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 + 3·7-s + 9-s − 4·11-s − 3·13-s + 5·19-s − 3·21-s + 6·23-s − 27-s + 29-s − 5·31-s + 4·33-s + 6·37-s + 3·39-s − 5·41-s − 43-s − 6·47-s + 2·49-s + 6·53-s − 5·57-s − 14·59-s + 61-s + 3·63-s − 11·67-s − 6·69-s − 11·73-s − 12·77-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.13·7-s + 1/3·9-s − 1.20·11-s − 0.832·13-s + 1.14·19-s − 0.654·21-s + 1.25·23-s − 0.192·27-s + 0.185·29-s − 0.898·31-s + 0.696·33-s + 0.986·37-s + 0.480·39-s − 0.780·41-s − 0.152·43-s − 0.875·47-s + 2/7·49-s + 0.824·53-s − 0.662·57-s − 1.82·59-s + 0.128·61-s + 0.377·63-s − 1.34·67-s − 0.722·69-s − 1.28·73-s − 1.36·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51600\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(412.028\)
Root analytic conductor: \(20.2984\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{51600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 51600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.608088497\)
\(L(\frac12)\) \(\approx\) \(1.608088497\)
\(L(\frac{3}{2})\) 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 \( 1 \)
43 \( 1 + T \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−14.60238318333469, −13.97882117651007, −13.33670904252591, −13.04431918270488, −12.31764309426621, −11.93443376197159, −11.25367587948935, −11.08001350929790, −10.38357724012219, −9.972862153312453, −9.318527981030939, −8.746404071829201, −8.044292713197418, −7.512532734758588, −7.323099069781327, −6.529306422916025, −5.659899884602093, −5.357760386146284, −4.694400896765552, −4.560444906802801, −3.329619076794381, −2.871142132915010, −2.007084393595933, −1.361228704820085, −0.4729888290352610, 0.4729888290352610, 1.361228704820085, 2.007084393595933, 2.871142132915010, 3.329619076794381, 4.560444906802801, 4.694400896765552, 5.357760386146284, 5.659899884602093, 6.529306422916025, 7.323099069781327, 7.512532734758588, 8.044292713197418, 8.746404071829201, 9.318527981030939, 9.972862153312453, 10.38357724012219, 11.08001350929790, 11.25367587948935, 11.93443376197159, 12.31764309426621, 13.04431918270488, 13.33670904252591, 13.97882117651007, 14.60238318333469

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