Properties

Degree $2$
Conductor $10830$
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 − 12-s − 2·13-s + 2·14-s + 15-s + 16-s − 6·17-s + 18-s − 20-s − 2·21-s + 6·23-s − 24-s + 25-s − 2·26-s − 27-s + 2·28-s + 4·29-s + 30-s + 32-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 − 0.288·12-s − 0.554·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.223·20-s − 0.436·21-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.377·28-s + 0.742·29-s + 0.182·30-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10830\)    =    \(2 \cdot 3 \cdot 5 \cdot 19^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{10830} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10830,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.457563733\)
\(L(\frac12)\) \(\approx\) \(2.457563733\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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

−16.34354129465694, −15.87672574440883, −15.40786934430931, −14.72701134060816, −14.42131626329995, −13.53339245158761, −13.11674941350205, −12.42219784575820, −11.93285336220763, −11.35916451020652, −10.87833304004861, −10.45675953176315, −9.497438609302446, −8.767467256188853, −8.172719756180136, −7.282331913686833, −6.942564697946708, −6.232957196003574, −5.358216688559009, −4.794548506616312, −4.414412770011866, −3.523587114738642, −2.595077735860410, −1.800866135356497, −0.6716349815713708, 0.6716349815713708, 1.800866135356497, 2.595077735860410, 3.523587114738642, 4.414412770011866, 4.794548506616312, 5.358216688559009, 6.232957196003574, 6.942564697946708, 7.282331913686833, 8.172719756180136, 8.767467256188853, 9.497438609302446, 10.45675953176315, 10.87833304004861, 11.35916451020652, 11.93285336220763, 12.42219784575820, 13.11674941350205, 13.53339245158761, 14.42131626329995, 14.72701134060816, 15.40786934430931, 15.87672574440883, 16.34354129465694

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