Properties

Degree 2
Conductor $ 5 \cdot 11 $
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

\( d \)  =  \(2\)
\( N \)  =  \(55\)    =    \(5 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{55} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 55,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.02866$
$L(\frac12)$  $\approx$  $1.02866$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
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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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.87836070551548265134861946988, −14.22243129432120604396205611002, −13.19684320404801561766277475369, −12.23057998502845892155732184830, −10.82193865307292862841462120688, −9.365267708430525132762162914792, −8.248642768475844318802620430882, −6.16821463973303244938343944630, −5.09766059609940450216711100862, −3.25423179226253980082861279896, 3.25423179226253980082861279896, 5.09766059609940450216711100862, 6.16821463973303244938343944630, 8.248642768475844318802620430882, 9.365267708430525132762162914792, 10.82193865307292862841462120688, 12.23057998502845892155732184830, 13.19684320404801561766277475369, 14.22243129432120604396205611002, 14.87836070551548265134861946988

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