Properties

Degree $2$
Conductor $169050$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 4·11-s − 12-s − 2·13-s + 16-s − 6·17-s − 18-s − 4·19-s + 4·22-s + 23-s + 24-s + 2·26-s − 27-s − 2·29-s + 8·31-s − 32-s + 4·33-s + 6·34-s + 36-s − 6·37-s + 4·38-s + 2·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s − 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.852·22-s + 0.208·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.176·32-s + 0.696·33-s + 1.02·34-s + 1/6·36-s − 0.986·37-s + 0.648·38-s + 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 23\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{169050} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 169050,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
23 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
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

−13.20754908903891, −13.01136852458465, −12.50280002072884, −11.94407436851224, −11.46274141013878, −10.93979747713873, −10.69002174630884, −10.14790693474266, −9.785583922325666, −9.132084717986867, −8.676341125176895, −8.179938831924650, −7.728433784710038, −7.116689188203952, −6.745253085582372, −6.201214631686480, −5.706844504571190, −5.069610557954549, −4.545244218436726, −4.185951140819508, −3.168140246629301, −2.645355351489815, −2.131938111932348, −1.513822734572897, −0.5254979815180808, 0, 0.5254979815180808, 1.513822734572897, 2.131938111932348, 2.645355351489815, 3.168140246629301, 4.185951140819508, 4.545244218436726, 5.069610557954549, 5.706844504571190, 6.201214631686480, 6.745253085582372, 7.116689188203952, 7.728433784710038, 8.179938831924650, 8.676341125176895, 9.132084717986867, 9.785583922325666, 10.14790693474266, 10.69002174630884, 10.93979747713873, 11.46274141013878, 11.94407436851224, 12.50280002072884, 13.01136852458465, 13.20754908903891

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