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 + 2·11-s + 12-s + 4·13-s + 15-s + 16-s − 2·17-s − 18-s + 20-s − 2·22-s − 6·23-s − 24-s + 25-s − 4·26-s + 27-s − 2·29-s − 30-s − 4·31-s − 32-s + 2·33-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 + 0.603·11-s + 0.288·12-s + 1.10·13-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.223·20-s − 0.426·22-s − 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s − 0.371·29-s − 0.182·30-s − 0.718·31-s − 0.176·32-s + 0.348·33-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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−16.78555227231845, −16.16361084378784, −15.87351011557003, −15.13383610506065, −14.47301625300522, −14.09263313856943, −13.33144381310014, −12.95378462921164, −12.10810050931958, −11.49705085239945, −10.93422365791823, −10.23162005041588, −9.767768284610108, −8.985935800875854, −8.746639609294130, −8.019469852470047, −7.393018581297996, −6.551794994576727, −6.189556759449027, −5.355109437061781, −4.333269777120354, −3.625771107089134, −2.927193445575052, −1.795020020381136, −1.508525466877444, 0, 1.508525466877444, 1.795020020381136, 2.927193445575052, 3.625771107089134, 4.333269777120354, 5.355109437061781, 6.189556759449027, 6.551794994576727, 7.393018581297996, 8.019469852470047, 8.746639609294130, 8.985935800875854, 9.767768284610108, 10.23162005041588, 10.93422365791823, 11.49705085239945, 12.10810050931958, 12.95378462921164, 13.33144381310014, 14.09263313856943, 14.47301625300522, 15.13383610506065, 15.87351011557003, 16.16361084378784, 16.78555227231845

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