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 + 6·11-s + 12-s − 4·13-s + 2·14-s − 15-s + 16-s − 6·17-s + 18-s + 19-s − 20-s + 2·21-s + 6·22-s + 24-s + 25-s − 4·26-s + 27-s + 2·28-s − 30-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 + 1.80·11-s + 0.288·12-s − 1.10·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.436·21-s + 1.27·22-s + 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.377·28-s − 0.182·30-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

\(L(1)\) \(\approx\) \(2.743573337\)
\(L(\frac12)\) \(\approx\) \(2.743573337\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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

−19.90106748401099, −19.70989370753348, −18.80623438547011, −17.62021442666613, −17.11372325687380, −16.16900250307718, −15.08855541882152, −14.88998086968726, −14.04000754508399, −13.41508127523667, −12.24604235496151, −11.75895978425587, −11.01785871675732, −9.790475980902335, −8.928245156415331, −8.025411245427375, −7.057932410813147, −6.313329011023513, −4.694544831293584, −4.286509102208767, −2.977103805297286, −1.663664536715263, 1.663664536715263, 2.977103805297286, 4.286509102208767, 4.694544831293584, 6.313329011023513, 7.057932410813147, 8.025411245427375, 8.928245156415331, 9.790475980902335, 11.01785871675732, 11.75895978425587, 12.24604235496151, 13.41508127523667, 14.04000754508399, 14.88998086968726, 15.08855541882152, 16.16900250307718, 17.11372325687380, 17.62021442666613, 18.80623438547011, 19.70989370753348, 19.90106748401099

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