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 − 2·7-s + 8-s + 9-s − 10-s + 12-s − 2·13-s − 2·14-s − 15-s + 16-s − 2·17-s + 18-s − 20-s − 2·21-s − 2·23-s + 24-s + 25-s − 2·26-s + 27-s − 2·28-s + 4·29-s − 30-s + 4·31-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 − 0.485·17-s + 0.235·18-s − 0.223·20-s − 0.436·21-s − 0.417·23-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 + 0.718·31-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
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

−16.64958765318396, −15.93409588731427, −15.64634516925209, −15.15815687156379, −14.43836163739419, −13.93875282259751, −13.48574664821230, −12.75018308537213, −12.32989544974756, −11.86306598161525, −11.01872493579644, −10.45342749163034, −9.792926257371640, −9.197821194150904, −8.480815611115517, −7.799479773384784, −7.216664133383383, −6.552468213261359, −6.004378343046248, −5.045603615568553, −4.381389455191669, −3.811887938382089, −2.936835113298251, −2.538897995190344, −1.385691988967204, 0, 1.385691988967204, 2.538897995190344, 2.936835113298251, 3.811887938382089, 4.381389455191669, 5.045603615568553, 6.004378343046248, 6.552468213261359, 7.216664133383383, 7.799479773384784, 8.480815611115517, 9.197821194150904, 9.792926257371640, 10.45342749163034, 11.01872493579644, 11.86306598161525, 12.32989544974756, 12.75018308537213, 13.48574664821230, 13.93875282259751, 14.43836163739419, 15.15815687156379, 15.64634516925209, 15.93409588731427, 16.64958765318396

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