Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 11·7-s + 23·13-s − 37·19-s + 25·25-s − 46·31-s − 73·37-s − 22·43-s + 72·49-s + 47·61-s − 13·67-s + 143·73-s + 11·79-s + 253·91-s − 169·97-s − 157·103-s − 214·109-s + ⋯
L(s)  = 1  + 11/7·7-s + 1.76·13-s − 1.94·19-s + 25-s − 1.48·31-s − 1.97·37-s − 0.511·43-s + 1.46·49-s + 0.770·61-s − 0.194·67-s + 1.95·73-s + 0.139·79-s + 2.78·91-s − 1.74·97-s − 1.52·103-s − 1.96·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  $\chi_{108} (53, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 1)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.54717\)
\(L(\frac12)\)  \(\approx\)  \(1.54717\)
\(L(2)\)   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 \( ( 1 - p T )( 1 + p T ) \)
7 \( 1 - 11 T + p^{2} T^{2} \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( 1 - 23 T + p^{2} T^{2} \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( 1 + 37 T + p^{2} T^{2} \)
23 \( ( 1 - p T )( 1 + p T ) \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( 1 + 46 T + p^{2} T^{2} \)
37 \( 1 + 73 T + p^{2} T^{2} \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( 1 + 22 T + p^{2} T^{2} \)
47 \( ( 1 - p T )( 1 + p T ) \)
53 \( ( 1 - p T )( 1 + p T ) \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( 1 - 47 T + p^{2} T^{2} \)
67 \( 1 + 13 T + p^{2} T^{2} \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( 1 - 143 T + p^{2} T^{2} \)
79 \( 1 - 11 T + p^{2} T^{2} \)
83 \( ( 1 - p T )( 1 + p T ) \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( 1 + 169 T + p^{2} T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.53445154385831040337638587948, −12.42697619570053259749815883598, −11.06117180223364224220707158157, −10.74228228770574286361146919753, −8.788020066621735944813417079021, −8.247366710731016436595046125187, −6.70993427037605020851683552027, −5.29288132048050207955676289818, −3.95174373493091634596021260943, −1.71116836367403221573069884983, 1.71116836367403221573069884983, 3.95174373493091634596021260943, 5.29288132048050207955676289818, 6.70993427037605020851683552027, 8.247366710731016436595046125187, 8.788020066621735944813417079021, 10.74228228770574286361146919753, 11.06117180223364224220707158157, 12.42697619570053259749815883598, 13.53445154385831040337638587948

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