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 − 2·13-s − 14-s + 15-s − 16-s + 2·17-s − 18-s + 4·19-s − 20-s + 21-s + 22-s + 3·24-s + 25-s + 2·26-s + 27-s − 28-s + 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.554·13-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 + 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-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

\(L(1)\) \(\approx\) \(1.370990110\)
\(L(\frac12)\) \(\approx\) \(1.370990110\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−19.87116544884411, −19.03907329589075, −18.46750703166868, −17.92161791424124, −17.28718950877292, −16.58999470071214, −15.84649245468392, −14.88642093131461, −14.21831920630561, −13.70327974491215, −12.98845604557206, −12.13978932930728, −11.15659645222843, −10.10921945617417, −9.842109718141052, −8.998624816402473, −8.198753615984875, −7.676735050819465, −6.711700870249909, −5.345720997077469, −4.718521494645523, −3.499343741266876, −2.273402764615337, −1.017573221785083, 1.017573221785083, 2.273402764615337, 3.499343741266876, 4.718521494645523, 5.345720997077469, 6.711700870249909, 7.676735050819465, 8.198753615984875, 8.998624816402473, 9.842109718141052, 10.10921945617417, 11.15659645222843, 12.13978932930728, 12.98845604557206, 13.70327974491215, 14.21831920630561, 14.88642093131461, 15.84649245468392, 16.58999470071214, 17.28718950877292, 17.92161791424124, 18.46750703166868, 19.03907329589075, 19.87116544884411

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