Properties

Degree $2$
Conductor $10830$
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 + 5-s + 6-s + 4·7-s − 8-s + 9-s − 10-s − 4·11-s − 12-s + 2·13-s − 4·14-s − 15-s + 16-s − 2·17-s − 18-s + 20-s − 4·21-s + 4·22-s − 8·23-s + 24-s + 25-s − 2·26-s − 27-s + 4·28-s − 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.554·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.223·20-s − 0.872·21-s + 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.755·28-s − 1.11·29-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: \(1\)
Selberg data: \((2,\ 10830,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
19 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−16.85707142473609, −16.37541509265479, −15.74136217380417, −15.27910477674490, −14.56668082350043, −14.00850555783635, −13.37074341796087, −12.67019339241295, −12.13944616169929, −11.19243986554018, −11.04176328402114, −10.63739162000011, −9.736904103607131, −9.293711087487954, −8.398238545083112, −7.819898083458938, −7.604293743516443, −6.566089938448824, −5.765052320731550, −5.495193094443058, −4.569413592582838, −3.937472419953114, −2.564990439392074, −1.990778112021967, −1.197880834181287, 0, 1.197880834181287, 1.990778112021967, 2.564990439392074, 3.937472419953114, 4.569413592582838, 5.495193094443058, 5.765052320731550, 6.566089938448824, 7.604293743516443, 7.819898083458938, 8.398238545083112, 9.293711087487954, 9.736904103607131, 10.63739162000011, 11.04176328402114, 11.19243986554018, 12.13944616169929, 12.67019339241295, 13.37074341796087, 14.00850555783635, 14.56668082350043, 15.27910477674490, 15.74136217380417, 16.37541509265479, 16.85707142473609

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