Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9i·5-s + 5·7-s + 117i·11-s − 34·13-s + 450i·17-s − 64·19-s + 612i·23-s + 544·25-s + 1.06e3i·29-s − 697·31-s + 45i·35-s − 748·37-s + 684i·41-s + 2.61e3·43-s − 2.64e3i·47-s + ⋯
L(s)  = 1  + 0.359i·5-s + 0.102·7-s + 0.966i·11-s − 0.201·13-s + 1.55i·17-s − 0.177·19-s + 1.15i·23-s + 0.870·25-s + 1.26i·29-s − 0.725·31-s + 0.0367i·35-s − 0.546·37-s + 0.406i·41-s + 1.41·43-s − 1.19i·47-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (53, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.00366 + 1.00366i\)
\(L(\frac12)\)  \(\approx\)  \(1.00366 + 1.00366i\)
\(L(3)\)   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 \( 1 - 9iT - 625T^{2} \)
7 \( 1 - 5T + 2.40e3T^{2} \)
11 \( 1 - 117iT - 1.46e4T^{2} \)
13 \( 1 + 34T + 2.85e4T^{2} \)
17 \( 1 - 450iT - 8.35e4T^{2} \)
19 \( 1 + 64T + 1.30e5T^{2} \)
23 \( 1 - 612iT - 2.79e5T^{2} \)
29 \( 1 - 1.06e3iT - 7.07e5T^{2} \)
31 \( 1 + 697T + 9.23e5T^{2} \)
37 \( 1 + 748T + 1.87e6T^{2} \)
41 \( 1 - 684iT - 2.82e6T^{2} \)
43 \( 1 - 2.61e3T + 3.41e6T^{2} \)
47 \( 1 + 2.64e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.07e3iT - 7.89e6T^{2} \)
59 \( 1 + 5.81e3iT - 1.21e7T^{2} \)
61 \( 1 - 6.40e3T + 1.38e7T^{2} \)
67 \( 1 + 5.21e3T + 2.01e7T^{2} \)
71 \( 1 + 6.57e3iT - 2.54e7T^{2} \)
73 \( 1 + 4.51e3T + 2.83e7T^{2} \)
79 \( 1 - 7.50e3T + 3.89e7T^{2} \)
83 \( 1 + 5.48e3iT - 4.74e7T^{2} \)
89 \( 1 - 8.87e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.05e4T + 8.85e7T^{2} \)
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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

−13.09186791230056260538149114567, −12.33158767946432614357884831771, −11.03801273603851871844645563982, −10.16218332862058325924040514815, −8.957264686496322542703138345278, −7.65873495575248042996558133761, −6.58311622632324317087726498629, −5.12205023863242180547277956679, −3.61446601895199997040693208283, −1.81074364198028364053065974691, 0.64334920823501279540036670758, 2.74604489179466577702223320029, 4.47663938633016616264341739330, 5.76164196829385482110821900163, 7.15524687000509825256436040542, 8.441811738788672930349170847216, 9.379555639978207914235101469964, 10.70007531978504615614967744152, 11.67421060643095213104317458864, 12.73343990958904782252015097480

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