Properties

Degree $2$
Conductor $55$
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 − 4-s + 5-s − 3·8-s − 3·9-s + 10-s − 11-s + 2·13-s − 16-s + 6·17-s − 3·18-s − 4·19-s − 20-s − 22-s + 4·23-s + 25-s + 2·26-s + 6·29-s − 8·31-s + 5·32-s + 6·34-s + 3·36-s − 2·37-s − 4·38-s − 3·40-s + 2·41-s + 4·43-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.06·8-s − 9-s + 0.316·10-s − 0.301·11-s + 0.554·13-s − 1/4·16-s + 1.45·17-s − 0.707·18-s − 0.917·19-s − 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s + 1.11·29-s − 1.43·31-s + 0.883·32-s + 1.02·34-s + 1/2·36-s − 0.328·37-s − 0.648·38-s − 0.474·40-s + 0.312·41-s + 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{55} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.028669856\)
\(L(\frac12)\) \(\approx\) \(1.028669856\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
11 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
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

−19.48247052421658, −18.39714892418324, −17.47697076219009, −16.43846695136001, −14.87836070551548, −14.22243129432121, −13.19684320404802, −12.23057998502846, −10.82193865307293, −9.365267708430525, −8.248642768475844, −6.168214639733032, −5.097660596099405, −3.254231792262540, 3.254231792262540, 5.097660596099405, 6.168214639733032, 8.248642768475844, 9.365267708430525, 10.82193865307293, 12.23057998502846, 13.19684320404802, 14.22243129432121, 14.87836070551548, 16.43846695136001, 17.47697076219009, 18.39714892418324, 19.48247052421658

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