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 − 5·7-s − 8-s + 9-s − 10-s + 11-s − 12-s + 6·13-s + 5·14-s − 15-s + 16-s + 4·17-s − 18-s + 20-s + 5·21-s − 22-s + 7·23-s + 24-s + 25-s − 6·26-s − 27-s − 5·28-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 − 1.88·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s + 1.66·13-s + 1.33·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.223·20-s + 1.09·21-s − 0.213·22-s + 1.45·23-s + 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.944·28-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: \(0\)
Selberg data: \((2,\ 10830,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.320878186\)
\(L(\frac12)\) \(\approx\) \(1.320878186\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
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

−16.55259624116890, −16.18671175582212, −15.60395350579221, −15.10657874160144, −14.15192046610076, −13.50584766572207, −12.97648293231160, −12.58095293922750, −11.82664051350225, −11.22166547442214, −10.46752979082809, −10.19563776814222, −9.411798919632051, −9.096495047997554, −8.372329239193445, −7.485764186712259, −6.670626885092130, −6.385961039225069, −5.892375323147863, −5.098226646371983, −3.896573123325929, −3.299989283584259, −2.632580200596660, −1.260495023524845, −0.7147988577571983, 0.7147988577571983, 1.260495023524845, 2.632580200596660, 3.299989283584259, 3.896573123325929, 5.098226646371983, 5.892375323147863, 6.385961039225069, 6.670626885092130, 7.485764186712259, 8.372329239193445, 9.096495047997554, 9.411798919632051, 10.19563776814222, 10.46752979082809, 11.22166547442214, 11.82664051350225, 12.58095293922750, 12.97648293231160, 13.50584766572207, 14.15192046610076, 15.10657874160144, 15.60395350579221, 16.18671175582212, 16.55259624116890

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