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  + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s − 7-s + 9-s − 2·10-s − 11-s − 2·12-s − 2·13-s − 2·14-s + 15-s − 4·16-s − 3·17-s + 2·18-s − 5·19-s − 2·20-s + 21-s − 2·22-s + 3·23-s + 25-s − 4·26-s − 27-s − 2·28-s − 3·29-s + 2·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s − 0.377·7-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.577·12-s − 0.554·13-s − 0.534·14-s + 0.258·15-s − 16-s − 0.727·17-s + 0.471·18-s − 1.14·19-s − 0.447·20-s + 0.218·21-s − 0.426·22-s + 0.625·23-s + 1/5·25-s − 0.784·26-s − 0.192·27-s − 0.377·28-s − 0.557·29-s + 0.365·30-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 \( 1 - p T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−19.59131061510791, −18.98463918140276, −18.13845492809312, −17.34286717171713, −16.62028650764706, −15.85896288760263, −15.16772378862265, −14.82595688292001, −13.73338982464420, −13.18788723462766, −12.46475532788285, −12.06161029395146, −11.10234517905379, −10.60096921816531, −9.422001093043397, −8.563023038477538, −7.308634625832175, −6.648045171280322, −5.839247066070870, −4.946801442973115, −4.303768540476930, −3.359222192774800, −2.241418508673228, 0, 2.241418508673228, 3.359222192774800, 4.303768540476930, 4.946801442973115, 5.839247066070870, 6.648045171280322, 7.308634625832175, 8.563023038477538, 9.422001093043397, 10.60096921816531, 11.10234517905379, 12.06161029395146, 12.46475532788285, 13.18788723462766, 13.73338982464420, 14.82595688292001, 15.16772378862265, 15.85896288760263, 16.62028650764706, 17.34286717171713, 18.13845492809312, 18.98463918140276, 19.59131061510791

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