Properties

Degree $2$
Conductor $1155$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s + 5-s − 7-s + 9-s − 11-s − 2·12-s − 4·13-s + 15-s + 4·16-s − 5·17-s + 19-s − 2·20-s − 21-s − 5·23-s + 25-s + 27-s + 2·28-s + 3·29-s − 6·31-s − 33-s − 35-s − 2·36-s − 12·37-s − 4·39-s − 2·41-s + 13·43-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 1.10·13-s + 0.258·15-s + 16-s − 1.21·17-s + 0.229·19-s − 0.447·20-s − 0.218·21-s − 1.04·23-s + 1/5·25-s + 0.192·27-s + 0.377·28-s + 0.557·29-s − 1.07·31-s − 0.174·33-s − 0.169·35-s − 1/3·36-s − 1.97·37-s − 0.640·39-s − 0.312·41-s + 1.98·43-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: \(1\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + T \)
good2 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 13 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 5 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−19.42222795317702, −18.98534524543550, −17.89005404982980, −17.81437503040350, −16.90348618454452, −16.00799050333130, −15.31516200624249, −14.40252510526641, −14.00018513389488, −13.27603453666148, −12.66775192834081, −11.97957357459895, −10.61323501010636, −10.06672324566815, −9.248260740270119, −8.821984320090431, −7.828141859093910, −7.025866826610126, −5.899287907849677, −4.960887084896385, −4.164052538615034, −3.063473196723552, −1.937616765106156, 0, 1.937616765106156, 3.063473196723552, 4.164052538615034, 4.960887084896385, 5.899287907849677, 7.025866826610126, 7.828141859093910, 8.821984320090431, 9.248260740270119, 10.06672324566815, 10.61323501010636, 11.97957357459895, 12.66775192834081, 13.27603453666148, 14.00018513389488, 14.40252510526641, 15.31516200624249, 16.00799050333130, 16.90348618454452, 17.81437503040350, 17.89005404982980, 18.98534524543550, 19.42222795317702

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