Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 13-s − 19-s + 25-s + 2·31-s − 37-s + 2·43-s − 61-s − 67-s − 73-s − 79-s + 91-s − 97-s − 103-s + 2·109-s + ⋯
L(s)  = 1  − 7-s − 13-s − 19-s + 25-s + 2·31-s − 37-s + 2·43-s − 61-s − 67-s − 73-s − 79-s + 91-s − 97-s − 103-s + 2·109-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{108} (53, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.5402022062\)
\(L(\frac12)\)  \(\approx\)  \(0.5402022062\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

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

−13.92836271074950410633497392280, −12.81147910151262800003561171965, −12.11393391520275622911136162744, −10.66337417717924861137567235324, −9.764363290770792093036581350306, −8.645382261510086690609111542994, −7.20655230074699554971488628053, −6.14349374284521055741894837740, −4.54012115448883134462155601253, −2.82161410923512167788607608967, 2.82161410923512167788607608967, 4.54012115448883134462155601253, 6.14349374284521055741894837740, 7.20655230074699554971488628053, 8.645382261510086690609111542994, 9.764363290770792093036581350306, 10.66337417717924861137567235324, 12.11393391520275622911136162744, 12.81147910151262800003561171965, 13.92836271074950410633497392280

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