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 − 2·7-s − 8-s + 9-s − 10-s + 12-s − 6·13-s + 2·14-s + 15-s + 16-s + 8·17-s − 18-s + 20-s − 2·21-s − 4·23-s − 24-s + 25-s + 6·26-s + 27-s − 2·28-s − 2·29-s − 30-s + 2·31-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 − 1.66·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s + 1.94·17-s − 0.235·18-s + 0.223·20-s − 0.436·21-s − 0.834·23-s − 0.204·24-s + 1/5·25-s + 1.17·26-s + 0.192·27-s − 0.377·28-s − 0.371·29-s − 0.182·30-s + 0.359·31-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

\(L(1)\) \(\approx\) \(1.626168059\)
\(L(\frac12)\) \(\approx\) \(1.626168059\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.63035732823783, −16.00058404997017, −15.51144151701027, −14.73295041802118, −14.21348518344492, −13.98882415930402, −12.83097580647012, −12.50879463846963, −12.10067190702360, −11.20267735409001, −10.34141746269217, −9.965312545423086, −9.486667267037489, −9.136143587100986, −8.123747898386931, −7.512573005444289, −7.335359657596467, −6.198040468798824, −5.825443325091038, −4.899542082405419, −3.997074830499344, −3.039583306911809, −2.619917157961974, −1.707732035740075, −0.6342614732890374, 0.6342614732890374, 1.707732035740075, 2.619917157961974, 3.039583306911809, 3.997074830499344, 4.899542082405419, 5.825443325091038, 6.198040468798824, 7.335359657596467, 7.512573005444289, 8.123747898386931, 9.136143587100986, 9.486667267037489, 9.965312545423086, 10.34141746269217, 11.20267735409001, 12.10067190702360, 12.50879463846963, 12.83097580647012, 13.98882415930402, 14.21348518344492, 14.73295041802118, 15.51144151701027, 16.00058404997017, 16.63035732823783

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