Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.287 + 0.957i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.821 + 5.59i)2-s + (−30.6 + 9.19i)4-s + 8.15i·5-s + 63.0i·7-s + (−76.6 − 164. i)8-s + (−45.6 + 6.69i)10-s − 442.·11-s − 76.3·13-s + (−352. + 51.7i)14-s + (855. − 563. i)16-s − 643. i·17-s − 2.18e3i·19-s + (−74.9 − 250. i)20-s + (−363. − 2.47e3i)22-s − 2.73e3·23-s + ⋯
L(s)  = 1  + (0.145 + 0.989i)2-s + (−0.957 + 0.287i)4-s + 0.145i·5-s + 0.485i·7-s + (−0.423 − 0.906i)8-s + (−0.144 + 0.0211i)10-s − 1.10·11-s − 0.125·13-s + (−0.480 + 0.0705i)14-s + (0.835 − 0.550i)16-s − 0.540i·17-s − 1.39i·19-s + (−0.0419 − 0.139i)20-s + (−0.160 − 1.09i)22-s − 1.07·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.287 + 0.957i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ 0.287 + 0.957i)\)
\(L(3)\)  \(\approx\)  \(0.291469 - 0.216891i\)
\(L(\frac12)\)  \(\approx\)  \(0.291469 - 0.216891i\)
\(L(\frac{7}{2})\)   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 \( 1 + (-0.821 - 5.59i)T \)
3 \( 1 \)
good5 \( 1 - 8.15iT - 3.12e3T^{2} \)
7 \( 1 - 63.0iT - 1.68e4T^{2} \)
11 \( 1 + 442.T + 1.61e5T^{2} \)
13 \( 1 + 76.3T + 3.71e5T^{2} \)
17 \( 1 + 643. iT - 1.41e6T^{2} \)
19 \( 1 + 2.18e3iT - 2.47e6T^{2} \)
23 \( 1 + 2.73e3T + 6.43e6T^{2} \)
29 \( 1 + 1.50e3iT - 2.05e7T^{2} \)
31 \( 1 + 5.50e3iT - 2.86e7T^{2} \)
37 \( 1 + 4.82e3T + 6.93e7T^{2} \)
41 \( 1 - 1.09e4iT - 1.15e8T^{2} \)
43 \( 1 + 8.35e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.39e4T + 2.29e8T^{2} \)
53 \( 1 + 2.28e4iT - 4.18e8T^{2} \)
59 \( 1 + 4.83e4T + 7.14e8T^{2} \)
61 \( 1 - 5.10e3T + 8.44e8T^{2} \)
67 \( 1 - 3.73e4iT - 1.35e9T^{2} \)
71 \( 1 + 7.50e4T + 1.80e9T^{2} \)
73 \( 1 + 6.52e4T + 2.07e9T^{2} \)
79 \( 1 - 7.32e4iT - 3.07e9T^{2} \)
83 \( 1 + 6.52e4T + 3.93e9T^{2} \)
89 \( 1 + 9.20e3iT - 5.58e9T^{2} \)
97 \( 1 - 1.25e5T + 8.58e9T^{2} \)
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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

−12.88298058723591011946992996080, −11.64324948022896362440241330150, −10.19510855525732366113680085231, −9.069014215232798261700608450049, −8.004046209897939829077621753315, −6.93967237991142753612565572291, −5.66791644397259407882318292826, −4.61388191229714509083594842881, −2.79994953648595759278393695543, −0.13036777292092968409116174284, 1.60304599912431397563386432224, 3.21720840408391944515033280059, 4.55814994966356496398061287484, 5.82924293301247829492002238329, 7.71091522652835688910186813090, 8.782572172909747557521524072168, 10.23938149051712404902894630470, 10.61459075301398949434673082548, 12.08115575269090284596758989672, 12.77289003137173566962245761174

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