Properties

Degree $2$
Conductor $53550$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 7-s + 8-s − 4·11-s + 2·13-s − 14-s + 16-s + 17-s + 4·19-s − 4·22-s + 8·23-s + 2·26-s − 28-s − 6·29-s + 32-s + 34-s + 2·37-s + 4·38-s − 10·41-s + 4·43-s − 4·44-s + 8·46-s + 49-s + 2·52-s + 6·53-s − 56-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.20·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.917·19-s − 0.852·22-s + 1.66·23-s + 0.392·26-s − 0.188·28-s − 1.11·29-s + 0.176·32-s + 0.171·34-s + 0.328·37-s + 0.648·38-s − 1.56·41-s + 0.609·43-s − 0.603·44-s + 1.17·46-s + 1/7·49-s + 0.277·52-s + 0.824·53-s − 0.133·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(53550\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{53550} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 53550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.634800373\)
\(L(\frac12)\) \(\approx\) \(3.634800373\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
17 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.33467579887119, −13.89453966323142, −13.28334220505923, −13.00328779568950, −12.64928331092936, −11.91399393517697, −11.29482022295949, −11.07755437903584, −10.32333206321893, −9.963968053061162, −9.258831068584352, −8.734330677180255, −8.011407384165695, −7.572895061673066, −6.930985074576201, −6.553113776397495, −5.655851395540180, −5.287158810336325, −4.972130937733457, −3.936544876924875, −3.540090755288284, −2.859043606956638, −2.368883295761019, −1.404641706912393, −0.6009670475093881, 0.6009670475093881, 1.404641706912393, 2.368883295761019, 2.859043606956638, 3.540090755288284, 3.936544876924875, 4.972130937733457, 5.287158810336325, 5.655851395540180, 6.553113776397495, 6.930985074576201, 7.572895061673066, 8.011407384165695, 8.734330677180255, 9.258831068584352, 9.963968053061162, 10.32333206321893, 11.07755437903584, 11.29482022295949, 11.91399393517697, 12.64928331092936, 13.00328779568950, 13.28334220505923, 13.89453966323142, 14.33467579887119

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