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 − 7-s − 8-s + 9-s + 10-s + 2·11-s − 12-s + 3·13-s + 14-s + 15-s + 16-s + 4·17-s − 18-s − 20-s + 21-s − 2·22-s − 6·23-s + 24-s + 25-s − 3·26-s − 27-s − 28-s + 10·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s + 0.832·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.223·20-s + 0.218·21-s − 0.426·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.588·26-s − 0.192·27-s − 0.188·28-s + 1.85·29-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: \(0\)
Selberg data: \((2,\ 10830,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.128352673\)
\(L(\frac12)\) \(\approx\) \(1.128352673\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
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

−16.48149095128845, −16.14825185469101, −15.62800392601227, −14.97461803799862, −14.27783806102610, −13.70909004972800, −12.92696171929070, −12.23254642169094, −11.85269218342290, −11.37399997942315, −10.60885358459513, −10.08868003334767, −9.632678456349248, −8.798303948139290, −8.186000151554346, −7.748484047761635, −6.732367810849687, −6.479639588638896, −5.748554351471988, −4.951714637135148, −3.964820620781226, −3.490540152562773, −2.454844044767427, −1.344850939526273, −0.6334608580750001, 0.6334608580750001, 1.344850939526273, 2.454844044767427, 3.490540152562773, 3.964820620781226, 4.951714637135148, 5.748554351471988, 6.479639588638896, 6.732367810849687, 7.748484047761635, 8.186000151554346, 8.798303948139290, 9.632678456349248, 10.08868003334767, 10.60885358459513, 11.37399997942315, 11.85269218342290, 12.23254642169094, 12.92696171929070, 13.70909004972800, 14.27783806102610, 14.97461803799862, 15.62800392601227, 16.14825185469101, 16.48149095128845

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