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 − 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: \(1\)
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 \( 1 + T + p T^{2} \)
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

−16.75664994547372, −16.43554566958720, −15.47410988113275, −15.01421014025037, −14.69728665772738, −13.93327637858641, −13.40681863898472, −12.47841671109537, −12.30120809435260, −11.79077989824837, −11.15716220442066, −10.62028238710020, −9.813225325754132, −9.201168793804266, −8.691832002955382, −7.481490571648136, −7.067362620102841, −6.700124811829989, −5.807973100410090, −5.185225853846472, −4.536345117409298, −3.779593761926864, −3.291924525593791, −2.155371750626999, −1.256911846465467, 0, 1.256911846465467, 2.155371750626999, 3.291924525593791, 3.779593761926864, 4.536345117409298, 5.185225853846472, 5.807973100410090, 6.700124811829989, 7.067362620102841, 7.481490571648136, 8.691832002955382, 9.201168793804266, 9.813225325754132, 10.62028238710020, 11.15716220442066, 11.79077989824837, 12.30120809435260, 12.47841671109537, 13.40681863898472, 13.93327637858641, 14.69728665772738, 15.01421014025037, 15.47410988113275, 16.43554566958720, 16.75664994547372

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