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 + 3·7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 2·13-s − 3·14-s − 15-s + 16-s + 2·17-s − 18-s − 20-s + 3·21-s − 22-s + 23-s − 24-s + 25-s + 2·26-s + 27-s + 3·28-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 + 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.554·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.223·20-s + 0.654·21-s − 0.213·22-s + 0.208·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.566·28-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 \( 1 - 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−16.86127719662636, −16.33444324940232, −15.56346982263631, −15.15959844300378, −14.47807114219386, −14.29476656201405, −13.43018153986808, −12.69870080675718, −11.96582430323238, −11.68166055006001, −10.90850530752034, −10.38520567032846, −9.739865677043213, −9.007519216932497, −8.538751178049366, −7.980006881390608, −7.391443016731359, −6.967366255832157, −5.990184434818289, −5.086984331504731, −4.560524706642520, −3.585413461396830, −2.954261690715228, −1.870048763815578, −1.387156666629877, 0, 1.387156666629877, 1.870048763815578, 2.954261690715228, 3.585413461396830, 4.560524706642520, 5.086984331504731, 5.990184434818289, 6.967366255832157, 7.391443016731359, 7.980006881390608, 8.538751178049366, 9.007519216932497, 9.739865677043213, 10.38520567032846, 10.90850530752034, 11.68166055006001, 11.96582430323238, 12.69870080675718, 13.43018153986808, 14.29476656201405, 14.47807114219386, 15.15959844300378, 15.56346982263631, 16.33444324940232, 16.86127719662636

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