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 − 4·11-s − 12-s − 6·13-s − 2·14-s + 15-s + 16-s + 4·17-s − 18-s − 20-s − 2·21-s + 4·22-s + 24-s + 25-s + 6·26-s − 27-s + 2·28-s + 10·29-s − 30-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 − 1.20·11-s − 0.288·12-s − 1.66·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.223·20-s − 0.436·21-s + 0.852·22-s + 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s + 0.377·28-s + 1.85·29-s − 0.182·30-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

\(L(1)\) \(\approx\) \(0.6853599521\)
\(L(\frac12)\) \(\approx\) \(0.6853599521\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−16.52409938030606, −16.13811162552998, −15.46059850467103, −14.90519758923569, −14.49571192753148, −13.64991667992168, −12.92619122171817, −12.12605623556955, −11.95903268041916, −11.37016520367209, −10.45816294020744, −10.21715376841859, −9.743130917711546, −8.690874830768288, −8.094671529531466, −7.705258795021992, −7.084543343896848, −6.427274223469056, −5.338141181228820, −5.062325263458210, −4.366312282824531, −3.120300507957763, −2.514117484067411, −1.498850543722726, −0.4515716238594288, 0.4515716238594288, 1.498850543722726, 2.514117484067411, 3.120300507957763, 4.366312282824531, 5.062325263458210, 5.338141181228820, 6.427274223469056, 7.084543343896848, 7.705258795021992, 8.094671529531466, 8.690874830768288, 9.743130917711546, 10.21715376841859, 10.45816294020744, 11.37016520367209, 11.95903268041916, 12.12605623556955, 12.92619122171817, 13.64991667992168, 14.49571192753148, 14.90519758923569, 15.46059850467103, 16.13811162552998, 16.52409938030606

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