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 + 2·13-s − 2·14-s − 15-s + 16-s − 6·17-s − 18-s − 20-s + 2·21-s + 6·23-s − 24-s + 25-s − 2·26-s + 27-s + 2·28-s − 4·29-s + 30-s − 32-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 − 1.45·17-s − 0.235·18-s − 0.223·20-s + 0.436·21-s + 1.25·23-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 − 0.176·32-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

\(L(1)\) \(\approx\) \(1.846873187\)
\(L(\frac12)\) \(\approx\) \(1.846873187\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−16.61856771600269, −15.77620192104397, −15.52491715106253, −14.81136041297686, −14.52819031880499, −13.59945928024524, −13.10230063518914, −12.59072694855512, −11.56286395472933, −11.21481721968218, −10.89578537544265, −9.979052969687440, −9.323882792995080, −8.802431576693408, −8.229609354138632, −7.848686405890593, −6.931963175085582, −6.646717972144258, −5.565165346526244, −4.763375793034925, −4.090258205589947, −3.275653654416408, −2.415225343332932, −1.681407139682146, −0.6968427046011265, 0.6968427046011265, 1.681407139682146, 2.415225343332932, 3.275653654416408, 4.090258205589947, 4.763375793034925, 5.565165346526244, 6.646717972144258, 6.931963175085582, 7.848686405890593, 8.229609354138632, 8.802431576693408, 9.323882792995080, 9.979052969687440, 10.89578537544265, 11.21481721968218, 11.56286395472933, 12.59072694855512, 13.10230063518914, 13.59945928024524, 14.52819031880499, 14.81136041297686, 15.52491715106253, 15.77620192104397, 16.61856771600269

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