Properties

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

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 11-s − 12-s + 16-s + 2·17-s − 18-s − 22-s + 24-s − 25-s − 27-s − 32-s − 33-s − 2·34-s + 36-s − 2·41-s + 44-s − 48-s − 49-s + 50-s − 2·51-s + 54-s + 64-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 11-s − 12-s + 16-s + 2·17-s − 18-s − 22-s + 24-s − 25-s − 27-s − 32-s − 33-s − 2·34-s + 36-s − 2·41-s + 44-s − 48-s − 49-s + 50-s − 2·51-s + 54-s + 64-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{264} (131, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 264,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.4105270841$
$L(\frac12)$  $\approx$  $0.4105270841$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;11\}$, \(F_p\) is a polynomial of degree 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 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( ( 1 - T )^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 + T )^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T^{2} \)
67 \( ( 1 - T )^{2} \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T^{2} \)
83 \( ( 1 + T )^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 + T )^{2} \)
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

−11.88451989650797107904539781928, −11.30238767356822738718639092233, −10.09069153626929297335704447362, −9.675979487524854751586233347964, −8.294767804190940999998727958881, −7.27754440837644720701665491952, −6.33764073208760212040798631320, −5.38007425380953534303618242882, −3.60975030390953103469712926046, −1.44474556202835591116477007583, 1.44474556202835591116477007583, 3.60975030390953103469712926046, 5.38007425380953534303618242882, 6.33764073208760212040798631320, 7.27754440837644720701665491952, 8.294767804190940999998727958881, 9.675979487524854751586233347964, 10.09069153626929297335704447362, 11.30238767356822738718639092233, 11.88451989650797107904539781928

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