Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.250 + 0.968i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 1.11i)2-s + (−0.500 − 1.93i)4-s − 2.23i·5-s + 3.87i·7-s + (−2.59 − 1.11i)8-s + (−2.50 − 1.93i)10-s + 1.73·11-s + 2·13-s + (4.33 + 3.35i)14-s + (−3.5 + 1.93i)16-s + 4.47i·17-s + (−4.33 + 1.11i)20-s + (1.49 − 1.93i)22-s − 6.92·23-s + (1.73 − 2.23i)26-s + ⋯
L(s)  = 1  + (0.612 − 0.790i)2-s + (−0.250 − 0.968i)4-s − 0.999i·5-s + 1.46i·7-s + (−0.918 − 0.395i)8-s + (−0.790 − 0.612i)10-s + 0.522·11-s + 0.554·13-s + (1.15 + 0.896i)14-s + (−0.875 + 0.484i)16-s + 1.08i·17-s + (−0.968 + 0.250i)20-s + (0.319 − 0.412i)22-s − 1.44·23-s + (0.339 − 0.438i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.250 + 0.968i$
motivic weight  =  \(1\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 108,\ (\ :1/2),\ 0.250 + 0.968i)$
$L(1)$  $\approx$  $1.04493 - 0.809405i$
$L(\frac12)$  $\approx$  $1.04493 - 0.809405i$
$L(\frac{3}{2})$   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 \( 1 + (-0.866 + 1.11i)T \)
3 \( 1 \)
good5 \( 1 + 2.23iT - 5T^{2} \)
7 \( 1 - 3.87iT - 7T^{2} \)
11 \( 1 - 1.73T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 4.47iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 6.92T + 23T^{2} \)
29 \( 1 - 4.47iT - 29T^{2} \)
31 \( 1 + 3.87iT - 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 8.94iT - 41T^{2} \)
43 \( 1 + 7.74iT - 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 + 2.23iT - 53T^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 - 7.74iT - 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 5T + 73T^{2} \)
79 \( 1 + 7.74iT - 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 - 4.47iT - 89T^{2} \)
97 \( 1 - 11T + 97T^{2} \)
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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

−13.20348376924089824645762661193, −12.32055725947877312123762577566, −11.83281850511323705118450104508, −10.45531999192661812101557673859, −9.126766705999112045448516538687, −8.538632563341344523611852273051, −6.16214209540224520188128453925, −5.28446112990804827637455560910, −3.83899928371328895260030976330, −1.90617931312276687744763662728, 3.29784359895666266011650792700, 4.46019734629326974911101879499, 6.27211017914488672877122195994, 7.06768930669604369158877238253, 8.034496553199522573133678458453, 9.685870156326968567023733517772, 10.89541614104477725345707656231, 11.89439036180129266747417695937, 13.37846127571007189687100086758, 14.00477049043570272934452413266

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