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 + 6·11-s + 12-s + 5·13-s + 14-s − 15-s + 16-s − 18-s − 20-s − 21-s − 6·22-s + 6·23-s − 24-s + 25-s − 5·26-s + 27-s − 28-s + 6·29-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 − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s + 0.288·12-s + 1.38·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.223·20-s − 0.218·21-s − 1.27·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.980·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-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

\(L(1)\) \(\approx\) \(2.451702655\)
\(L(\frac12)\) \(\approx\) \(2.451702655\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−16.46845364747815, −16.00100234604119, −15.47835853311596, −14.90383992513259, −14.25427224088457, −13.84259539105867, −12.97788340424280, −12.54240307344790, −11.66926020405516, −11.35365438061613, −10.70986170433115, −9.944237018575117, −9.222605614943090, −8.982970336234498, −8.338788970520050, −7.732371084853932, −6.919078285997604, −6.421643957740092, −5.943241700041822, −4.570975010242444, −4.041945390825184, −3.272760778713735, −2.655063376838759, −1.323206996198386, −0.9488138736495322, 0.9488138736495322, 1.323206996198386, 2.655063376838759, 3.272760778713735, 4.041945390825184, 4.570975010242444, 5.943241700041822, 6.421643957740092, 6.919078285997604, 7.732371084853932, 8.338788970520050, 8.982970336234498, 9.222605614943090, 9.944237018575117, 10.70986170433115, 11.35365438061613, 11.66926020405516, 12.54240307344790, 12.97788340424280, 13.84259539105867, 14.25427224088457, 14.90383992513259, 15.47835853311596, 16.00100234604119, 16.46845364747815

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