Properties

Degree $2$
Conductor $570$
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 + 3-s + 4-s − 5-s + 6-s + 2·7-s + 8-s + 9-s − 10-s − 4·11-s + 12-s + 6·13-s + 2·14-s − 15-s + 16-s + 4·17-s + 18-s + 19-s − 20-s + 2·21-s − 4·22-s + 24-s + 25-s + 6·26-s + 27-s + 2·28-s − 10·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s + 1.66·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.436·21-s − 0.852·22-s + 0.204·24-s + 1/5·25-s + 1.17·26-s + 0.192·27-s + 0.377·28-s − 1.85·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{570} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.704191420\)
\(L(\frac12)\) \(\approx\) \(2.704191420\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 + T \)
19 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−19.34989849565202, −18.50189156999481, −18.16608720197880, −16.83921778298712, −16.01430440849869, −15.49989367885674, −14.69280502825453, −14.04375005780781, −13.17827389300044, −12.68647963126563, −11.43201859914607, −11.01677174780752, −10.00807079307099, −8.745310200714098, −7.948201996751845, −7.375754243568286, −5.953390006578408, −5.116877739513923, −3.936437289304420, −3.110638060568703, −1.630189689785315, 1.630189689785315, 3.110638060568703, 3.936437289304420, 5.116877739513923, 5.953390006578408, 7.375754243568286, 7.948201996751845, 8.745310200714098, 10.00807079307099, 11.01677174780752, 11.43201859914607, 12.68647963126563, 13.17827389300044, 14.04375005780781, 14.69280502825453, 15.49989367885674, 16.01430440849869, 16.83921778298712, 18.16608720197880, 18.50189156999481, 19.34989849565202

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