Properties

Degree 2
Conductor $ 3^{3} \cdot 5 $
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 + 5-s + 8-s − 10-s − 16-s − 17-s − 19-s − 23-s + 25-s − 31-s + 34-s + 38-s + 40-s + 46-s + 2·47-s + 49-s − 50-s − 53-s − 61-s + 62-s + 64-s − 79-s − 80-s − 83-s − 85-s − 2·94-s − 95-s + ⋯
L(s)  = 1  − 2-s + 5-s + 8-s − 10-s − 16-s − 17-s − 19-s − 23-s + 25-s − 31-s + 34-s + 38-s + 40-s + 46-s + 2·47-s + 49-s − 50-s − 53-s − 61-s + 62-s + 64-s − 79-s − 80-s − 83-s − 85-s − 2·94-s − 95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(135\)    =    \(3^{3} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{135} (134, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 135,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.4163687655\)
\(L(\frac12)\)  \(\approx\)  \(0.4163687655\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
good2 \( 1 + T + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )^{2} \)
53 \( 1 + T + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T + T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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

−13.51579545588941529611973298262, −12.59449163938264305142168973101, −10.98652682515726003864537765386, −10.25584206721761290089387432829, −9.253566177983878475275416700587, −8.548308440781228384134559841616, −7.19935214621441948680617539145, −5.91615283308312356244034077336, −4.40323847289512931191199822637, −2.04028622872293924908533618653, 2.04028622872293924908533618653, 4.40323847289512931191199822637, 5.91615283308312356244034077336, 7.19935214621441948680617539145, 8.548308440781228384134559841616, 9.253566177983878475275416700587, 10.25584206721761290089387432829, 10.98652682515726003864537765386, 12.59449163938264305142168973101, 13.51579545588941529611973298262

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