Properties

Degree $2$
Conductor $10830$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

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 + 12-s + 6·13-s − 4·14-s + 15-s + 16-s + 2·17-s − 18-s + 20-s + 4·21-s + 8·23-s − 24-s + 25-s − 6·26-s + 27-s + 4·28-s − 2·29-s − 30-s + 8·31-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 1.66·13-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 + 1.66·23-s − 0.204·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s + 0.755·28-s − 0.371·29-s − 0.182·30-s + 1.43·31-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: \(0\)
Selberg data: \((2,\ 10830,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.307619830\)
\(L(\frac12)\) \(\approx\) \(3.307619830\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−16.64839821420355, −15.93505133730800, −15.28897002965807, −14.95030520009353, −14.27820076564936, −13.61434283424702, −13.38230758749024, −12.41630693717189, −11.71761049143230, −11.17403868053337, −10.68407294689775, −10.13074683523724, −9.323521201346514, −8.722811069575623, −8.256227046676696, −7.968668558876545, −6.844459175849321, −6.621225733310573, −5.337999764838123, −5.145804313711865, −3.974957039066808, −3.291909627967540, −2.356403681700318, −1.478212599544676, −1.079645573244478, 1.079645573244478, 1.478212599544676, 2.356403681700318, 3.291909627967540, 3.974957039066808, 5.145804313711865, 5.337999764838123, 6.621225733310573, 6.844459175849321, 7.968668558876545, 8.256227046676696, 8.722811069575623, 9.323521201346514, 10.13074683523724, 10.68407294689775, 11.17403868053337, 11.71761049143230, 12.41630693717189, 13.38230758749024, 13.61434283424702, 14.27820076564936, 14.95030520009353, 15.28897002965807, 15.93505133730800, 16.64839821420355

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