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 − 2·7-s − 8-s + 9-s − 10-s + 2·11-s − 12-s − 4·13-s + 2·14-s − 15-s + 16-s − 2·17-s − 18-s + 20-s + 2·21-s − 2·22-s + 4·23-s + 24-s + 25-s + 4·26-s − 27-s − 2·28-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 + 0.603·11-s − 0.288·12-s − 1.10·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.223·20-s + 0.436·21-s − 0.426·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.377·28-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−16.79371246676279, −16.56169637926125, −15.70662250540592, −15.32981010348052, −14.58716600571997, −14.02035210333135, −13.16041739097285, −12.80043319141476, −12.00336045109163, −11.68677365818428, −10.88288540964571, −10.29942139782372, −9.750083997488944, −9.356109714978283, −8.677936623662044, −7.902381339968528, −7.076342073825572, −6.605412639939142, −6.220396424877922, −5.209268817453299, −4.726665771852384, −3.646210230617424, −2.823984488366728, −2.021379251290026, −1.016222731653324, 0, 1.016222731653324, 2.021379251290026, 2.823984488366728, 3.646210230617424, 4.726665771852384, 5.209268817453299, 6.220396424877922, 6.605412639939142, 7.076342073825572, 7.902381339968528, 8.677936623662044, 9.356109714978283, 9.750083997488944, 10.29942139782372, 10.88288540964571, 11.68677365818428, 12.00336045109163, 12.80043319141476, 13.16041739097285, 14.02035210333135, 14.58716600571997, 15.32981010348052, 15.70662250540592, 16.56169637926125, 16.79371246676279

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