Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 25·7-s − 427·13-s − 1.71e3·19-s − 3.12e3·25-s − 1.03e4·31-s − 6.66e3·37-s − 3.35e3·43-s − 1.61e4·49-s + 5.69e4·61-s − 3.79e4·67-s + 7.95e4·73-s + 9.08e4·79-s + 1.06e4·91-s + 1.77e5·97-s − 2.11e5·103-s + 1.14e5·109-s + ⋯
L(s)  = 1  − 0.192·7-s − 0.700·13-s − 1.08·19-s − 25-s − 1.92·31-s − 0.799·37-s − 0.276·43-s − 0.962·49-s + 1.95·61-s − 1.03·67-s + 1.74·73-s + 1.63·79-s + 0.135·91-s + 1.91·97-s − 1.96·103-s + 0.922·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(5\)
character  :  $\chi_{108} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ -1)\)
\(L(3)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + p^{5} T^{2} \)
7 \( 1 + 25 T + p^{5} T^{2} \)
11 \( 1 + p^{5} T^{2} \)
13 \( 1 + 427 T + p^{5} T^{2} \)
17 \( 1 + p^{5} T^{2} \)
19 \( 1 + 1711 T + p^{5} T^{2} \)
23 \( 1 + p^{5} T^{2} \)
29 \( 1 + p^{5} T^{2} \)
31 \( 1 + 10324 T + p^{5} T^{2} \)
37 \( 1 + 6661 T + p^{5} T^{2} \)
41 \( 1 + p^{5} T^{2} \)
43 \( 1 + 3352 T + p^{5} T^{2} \)
47 \( 1 + p^{5} T^{2} \)
53 \( 1 + p^{5} T^{2} \)
59 \( 1 + p^{5} T^{2} \)
61 \( 1 - 56927 T + p^{5} T^{2} \)
67 \( 1 + 37939 T + p^{5} T^{2} \)
71 \( 1 + p^{5} T^{2} \)
73 \( 1 - 79577 T + p^{5} T^{2} \)
79 \( 1 - 90857 T + p^{5} T^{2} \)
83 \( 1 + p^{5} T^{2} \)
89 \( 1 + p^{5} T^{2} \)
97 \( 1 - 177725 T + p^{5} T^{2} \)
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

−12.32237201020811695553306360796, −11.17158952591131540748894685748, −10.08755192862193851674052925045, −9.026219897673102285381807836202, −7.77397120410345652122810585011, −6.58516209533226628464575787824, −5.23407411169190455619298184721, −3.76291998709216436997159381084, −2.07232651685692341885779650387, 0, 2.07232651685692341885779650387, 3.76291998709216436997159381084, 5.23407411169190455619298184721, 6.58516209533226628464575787824, 7.77397120410345652122810585011, 9.026219897673102285381807836202, 10.08755192862193851674052925045, 11.17158952591131540748894685748, 12.32237201020811695553306360796

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