Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s − 11-s + 12-s − 4·13-s − 2·14-s + 16-s − 6·17-s − 18-s − 4·19-s + 2·21-s + 22-s + 6·23-s − 24-s − 5·25-s + 4·26-s + 27-s + 2·28-s + 6·29-s + 8·31-s − 32-s − 33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.436·21-s + 0.213·22-s + 1.25·23-s − 0.204·24-s − 25-s + 0.784·26-s + 0.192·27-s + 0.377·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.174·33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(66\)    =    \(2 \cdot 3 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{66} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 66,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7970914573$
$L(\frac12)$  $\approx$  $0.7970914573$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 - T \)
11 \( 1 + T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.58862248742375, −19.21427628733258, −17.70972806364377, −17.34325284037646, −15.77693353801864, −15.02818417573081, −13.92129713751308, −12.61238129145389, −11.31231841896419, −10.21971960261641, −8.958855021658912, −8.013104010237497, −6.775065655590044, −4.693334713327062, −2.375450913572559, 2.375450913572559, 4.693334713327062, 6.775065655590044, 8.013104010237497, 8.958855021658912, 10.21971960261641, 11.31231841896419, 12.61238129145389, 13.92129713751308, 15.02818417573081, 15.77693353801864, 17.34325284037646, 17.70972806364377, 19.21427628733258, 19.58862248742375

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