Properties

Degree $2$
Conductor $429$
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 + 3·8-s + 9-s + 11-s + 12-s + 13-s − 16-s − 4·17-s − 18-s − 8·19-s − 22-s − 3·24-s − 5·25-s − 26-s − 27-s + 4·29-s − 6·31-s − 5·32-s − 33-s + 4·34-s − 36-s − 6·37-s + 8·38-s − 39-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 0.277·13-s − 1/4·16-s − 0.970·17-s − 0.235·18-s − 1.83·19-s − 0.213·22-s − 0.612·24-s − 25-s − 0.196·26-s − 0.192·27-s + 0.742·29-s − 1.07·31-s − 0.883·32-s − 0.174·33-s + 0.685·34-s − 1/6·36-s − 0.986·37-s + 1.29·38-s − 0.160·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{429} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−19.44353012295296, −18.75271763573406, −17.86871299465849, −17.46462513185862, −16.74021811711282, −15.94753031987500, −15.01349755365290, −14.03777792898784, −13.16727004647310, −12.51336226771828, −11.29252528487813, −10.68195930442135, −9.805924573894417, −8.882281902623244, −8.198333501616002, −7.006668413057594, −6.077446718710783, −4.786795566950740, −3.939910762033627, −1.825715196240894, 0, 1.825715196240894, 3.939910762033627, 4.786795566950740, 6.077446718710783, 7.006668413057594, 8.198333501616002, 8.882281902623244, 9.805924573894417, 10.68195930442135, 11.29252528487813, 12.51336226771828, 13.16727004647310, 14.03777792898784, 15.01349755365290, 15.94753031987500, 16.74021811711282, 17.46462513185862, 17.86871299465849, 18.75271763573406, 19.44353012295296

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