Properties

Degree 2
Conductor $ 2^{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  − 3-s + 2·5-s + 2·7-s + 9-s − 11-s + 6·13-s − 2·15-s − 4·17-s − 2·19-s − 2·21-s − 8·23-s − 25-s − 27-s + 33-s + 4·35-s − 6·37-s − 6·39-s + 10·43-s + 2·45-s − 3·49-s + 4·51-s + 14·53-s − 2·55-s + 2·57-s − 12·59-s − 14·61-s + 2·63-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s − 0.516·15-s − 0.970·17-s − 0.458·19-s − 0.436·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 0.174·33-s + 0.676·35-s − 0.986·37-s − 0.960·39-s + 1.52·43-s + 0.298·45-s − 3/7·49-s + 0.560·51-s + 1.92·53-s − 0.269·55-s + 0.264·57-s − 1.56·59-s − 1.79·61-s + 0.251·63-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{132} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 132,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.094444217$
$L(\frac12)$  $\approx$  $1.094444217$
$L(\frac{3}{2})$   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 \( 1 \)
3 \( 1 + T \)
11 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−19.83809181880179, −18.41952846398506, −17.97352641832613, −17.26974943259681, −16.13323626875807, −15.36474788823861, −13.95654848901191, −13.45826335615790, −12.22139625462217, −11.07464839362546, −10.44502381868694, −9.106296123632975, −8.017674089261979, −6.418936276107659, −5.628608574786801, −4.180205639519794, −1.867850392490877, 1.867850392490877, 4.180205639519794, 5.628608574786801, 6.418936276107659, 8.017674089261979, 9.106296123632975, 10.44502381868694, 11.07464839362546, 12.22139625462217, 13.45826335615790, 13.95654848901191, 15.36474788823861, 16.13323626875807, 17.26974943259681, 17.97352641832613, 18.41952846398506, 19.83809181880179

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