Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
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  + 5·7-s − 7·13-s − 19-s − 5·25-s − 4·31-s − 37-s + 8·43-s + 18·49-s − 13·61-s + 11·67-s + 17·73-s − 13·79-s − 35·91-s + 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.88·7-s − 1.94·13-s − 0.229·19-s − 25-s − 0.718·31-s − 0.164·37-s + 1.21·43-s + 18/7·49-s − 1.66·61-s + 1.34·67-s + 1.98·73-s − 1.46·79-s − 3.66·91-s + 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{108} (1, \cdot )$
Sato-Tate  :  $N(\mathrm{U}(1))$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 108,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.112912674$
$L(\frac12)$  $\approx$  $1.112912674$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 5 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 17 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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

−19.50109065340506, −18.34367265706647, −17.45737388576188, −16.96902712966989, −15.43122998493178, −14.61655121900918, −14.00533773800539, −12.48081907715317, −11.64280365407061, −10.67338269741491, −9.420824192029041, −8.079851394884848, −7.313463101963462, −5.416195955808154, −4.441949961306198, −2.124962109282895, 2.124962109282895, 4.441949961306198, 5.416195955808154, 7.313463101963462, 8.079851394884848, 9.420824192029041, 10.67338269741491, 11.64280365407061, 12.48081907715317, 14.00533773800539, 14.61655121900918, 15.43122998493178, 16.96902712966989, 17.45737388576188, 18.34367265706647, 19.50109065340506

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