Properties

Degree $2$
Conductor $1155$
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 + 7-s − 3·8-s + 9-s + 10-s − 11-s − 12-s + 6·13-s + 14-s + 15-s − 16-s + 2·17-s + 18-s + 4·19-s − 20-s + 21-s − 22-s − 4·23-s − 3·24-s + 25-s + 6·26-s + 27-s − 28-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 − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s + 1.66·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s − 0.213·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s + 1.17·26-s + 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1155} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.824482658\)
\(L(\frac12)\) \(\approx\) \(2.824482658\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−18.95846566513185, −18.55410690831292, −17.90482287072486, −17.33906983756548, −16.07820925104838, −15.72328021839119, −14.75121507171002, −14.10348112203972, −13.68753137609328, −13.11304189990101, −12.30169166835495, −11.48435481127724, −10.51328421423159, −9.698646968210728, −8.909303336785777, −8.304701526069343, −7.400205455970474, −6.054311972332621, −5.616192372994298, −4.468629861736962, −3.686973633746718, −2.757410293174219, −1.262564673649024, 1.262564673649024, 2.757410293174219, 3.686973633746718, 4.468629861736962, 5.616192372994298, 6.054311972332621, 7.400205455970474, 8.304701526069343, 8.909303336785777, 9.698646968210728, 10.51328421423159, 11.48435481127724, 12.30169166835495, 13.11304189990101, 13.68753137609328, 14.10348112203972, 14.75121507171002, 15.72328021839119, 16.07820925104838, 17.33906983756548, 17.90482287072486, 18.55410690831292, 18.95846566513185

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