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 − 4·7-s − 8-s + 9-s + 10-s + 6·11-s + 12-s − 4·13-s + 4·14-s − 15-s + 16-s + 6·17-s − 18-s − 20-s − 4·21-s − 6·22-s + 6·23-s − 24-s + 25-s + 4·26-s + 27-s − 4·28-s + 2·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s + 0.288·12-s − 1.10·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.223·20-s − 0.872·21-s − 1.27·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s − 0.755·28-s + 0.371·29-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

\(L(1)\) \(\approx\) \(1.494429605\)
\(L(\frac12)\) \(\approx\) \(1.494429605\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−16.61837087870359, −16.12577650065411, −15.36066154760472, −14.88207672212260, −14.37594328184876, −13.80473700499483, −12.91290838304143, −12.31723132736317, −12.08806585016795, −11.35493158145793, −10.39961580490658, −9.972872108224414, −9.321079340493947, −9.091149335773776, −8.369683615595407, −7.352177422288140, −7.191588509536596, −6.486651519843896, −5.815705076669247, −4.761028105706978, −3.824541841713489, −3.275251286133736, −2.730631676664387, −1.512986528536166, −0.6443895030195491, 0.6443895030195491, 1.512986528536166, 2.730631676664387, 3.275251286133736, 3.824541841713489, 4.761028105706978, 5.815705076669247, 6.486651519843896, 7.191588509536596, 7.352177422288140, 8.369683615595407, 9.091149335773776, 9.321079340493947, 9.972872108224414, 10.39961580490658, 11.35493158145793, 12.08806585016795, 12.31723132736317, 12.91290838304143, 13.80473700499483, 14.37594328184876, 14.88207672212260, 15.36066154760472, 16.12577650065411, 16.61837087870359

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