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 − 2·13-s + 2·15-s + 4·17-s − 6·19-s − 2·21-s − 25-s + 27-s − 8·29-s − 8·31-s + 33-s − 4·35-s + 10·37-s − 2·39-s + 8·41-s − 2·43-s + 2·45-s − 8·47-s − 3·49-s + 4·51-s − 2·53-s + 2·55-s − 6·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.516·15-s + 0.970·17-s − 1.37·19-s − 0.436·21-s − 1/5·25-s + 0.192·27-s − 1.48·29-s − 1.43·31-s + 0.174·33-s − 0.676·35-s + 1.64·37-s − 0.320·39-s + 1.24·41-s − 0.304·43-s + 0.298·45-s − 1.16·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s + 0.269·55-s − 0.794·57-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  =  $(2,\ 132,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.328164578$
$L(\frac12)$  $\approx$  $1.328164578$
$L(\frac{3}{2})$   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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
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

−19.64799819475256, −19.00259728064606, −18.00940115907657, −16.94365587312752, −16.30509759495024, −14.87164667309865, −14.40625598696955, −13.10795468637364, −12.70566797794869, −11.13297096888405, −9.796917351475792, −9.413820553443584, −7.996057650348538, −6.704536431931973, −5.572735214177054, −3.792942356415599, −2.212521994355865, 2.212521994355865, 3.792942356415599, 5.572735214177054, 6.704536431931973, 7.996057650348538, 9.413820553443584, 9.796917351475792, 11.13297096888405, 12.70566797794869, 13.10795468637364, 14.40625598696955, 14.87164667309865, 16.30509759495024, 16.94365587312752, 18.00940115907657, 19.00259728064606, 19.64799819475256

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