Properties

Degree $2$
Conductor $429$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 4-s − 2·5-s − 6-s + 3·8-s + 9-s + 2·10-s − 11-s − 12-s + 13-s − 2·15-s − 16-s − 6·17-s − 18-s − 4·19-s + 2·20-s + 22-s − 8·23-s + 3·24-s − 25-s − 26-s + 27-s − 10·29-s + 2·30-s − 5·32-s − 33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.516·15-s − 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.213·22-s − 1.66·23-s + 0.612·24-s − 1/5·25-s − 0.196·26-s + 0.192·27-s − 1.85·29-s + 0.365·30-s − 0.883·32-s − 0.174·33-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 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.45532185767595, −18.94186042774100, −18.08490865143435, −17.54145995129915, −16.43410236352671, −15.80242451815289, −15.00360703420439, −14.11089306860001, −13.25148953479724, −12.61604856691270, −11.31119367581499, −10.67556634305418, −9.542919498336760, −8.901146201159458, −7.971709728009125, −7.550626730061059, −6.098406672733796, −4.442373125876683, −3.896412246855618, −2.096398229324023, 0, 2.096398229324023, 3.896412246855618, 4.442373125876683, 6.098406672733796, 7.550626730061059, 7.971709728009125, 8.901146201159458, 9.542919498336760, 10.67556634305418, 11.31119367581499, 12.61604856691270, 13.25148953479724, 14.11089306860001, 15.00360703420439, 15.80242451815289, 16.43410236352671, 17.54145995129915, 18.08490865143435, 18.94186042774100, 19.45532185767595

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