Properties

Degree $2$
Conductor $10830$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 4·11-s + 12-s − 2·13-s − 15-s + 16-s + 2·17-s − 18-s − 20-s − 4·22-s + 4·23-s − 24-s + 25-s + 2·26-s + 27-s − 6·29-s + 30-s − 4·31-s − 32-s + 4·33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.223·20-s − 0.852·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 1.11·29-s + 0.182·30-s − 0.718·31-s − 0.176·32-s + 0.696·33-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: \(1\)
Selberg data: \((2,\ 10830,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−16.65450072012401, −16.54164530518193, −15.66065650396600, −14.96646784041333, −14.68282453663971, −14.25592233938981, −13.20399995956973, −12.87024632486520, −12.06852350891394, −11.43243162470136, −11.20486331682982, −10.09803524436512, −9.775796203638255, −9.118998183727666, −8.598117045380319, −7.983722896797747, −7.295996723815246, −6.872499619262520, −6.127082053354310, −5.176024551567363, −4.414674891088173, −3.503837191665468, −3.090830565693202, −1.923387144643872, −1.286383262773349, 0, 1.286383262773349, 1.923387144643872, 3.090830565693202, 3.503837191665468, 4.414674891088173, 5.176024551567363, 6.127082053354310, 6.872499619262520, 7.295996723815246, 7.983722896797747, 8.598117045380319, 9.118998183727666, 9.775796203638255, 10.09803524436512, 11.20486331682982, 11.43243162470136, 12.06852350891394, 12.87024632486520, 13.20399995956973, 14.25592233938981, 14.68282453663971, 14.96646784041333, 15.66065650396600, 16.54164530518193, 16.65450072012401

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