Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.812 + 0.582i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 + 5.38i)2-s + (−25.9 + 18.6i)4-s + 91.5i·5-s + 158. i·7-s + (−145. − 107. i)8-s + (−493 + 158. i)10-s + 580.·11-s − 166·13-s + (−853. + 274. i)14-s + (327. − 970. i)16-s − 829. i·17-s + 671. i·19-s + (−1.70e3 − 2.38e3i)20-s + (1.00e3 + 3.12e3i)22-s − 3.85e3·23-s + ⋯
L(s)  = 1  + (0.306 + 0.951i)2-s + (−0.812 + 0.582i)4-s + 1.63i·5-s + 1.22i·7-s + (−0.803 − 0.594i)8-s + (−1.55 + 0.501i)10-s + 1.44·11-s − 0.272·13-s + (−1.16 + 0.374i)14-s + (0.320 − 0.947i)16-s − 0.695i·17-s + 0.426i·19-s + (−0.954 − 1.33i)20-s + (0.442 + 1.37i)22-s − 1.52·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.582i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.812 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.812 + 0.582i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ -0.812 + 0.582i)\)
\(L(3)\)  \(\approx\)  \(0.497536 - 1.54690i\)
\(L(\frac12)\)  \(\approx\)  \(0.497536 - 1.54690i\)
\(L(\frac{7}{2})\)   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 \( 1 + (-1.73 - 5.38i)T \)
3 \( 1 \)
good5 \( 1 - 91.5iT - 3.12e3T^{2} \)
7 \( 1 - 158. iT - 1.68e4T^{2} \)
11 \( 1 - 580.T + 1.61e5T^{2} \)
13 \( 1 + 166T + 3.71e5T^{2} \)
17 \( 1 + 829. iT - 1.41e6T^{2} \)
19 \( 1 - 671. iT - 2.47e6T^{2} \)
23 \( 1 + 3.85e3T + 6.43e6T^{2} \)
29 \( 1 + 3.41e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.33e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.53e4T + 6.93e7T^{2} \)
41 \( 1 + 1.01e4iT - 1.15e8T^{2} \)
43 \( 1 + 3.00e3iT - 1.47e8T^{2} \)
47 \( 1 - 7.07e3T + 2.29e8T^{2} \)
53 \( 1 - 1.72e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.84e4T + 7.14e8T^{2} \)
61 \( 1 + 5.31e4T + 8.44e8T^{2} \)
67 \( 1 - 4.10e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.68e4T + 1.80e9T^{2} \)
73 \( 1 + 3.07e4T + 2.07e9T^{2} \)
79 \( 1 - 7.01e4iT - 3.07e9T^{2} \)
83 \( 1 + 1.14e4T + 3.93e9T^{2} \)
89 \( 1 + 6.86e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.37e4T + 8.58e9T^{2} \)
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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

−13.92962088518035842269674556136, −12.23675247205873586500800016306, −11.59683044887002481692665274679, −9.972382748693880101890060608176, −8.968575089485689946828740757456, −7.61570506982922370239171693455, −6.54084307009892782626923672978, −5.81299978084502266307040812790, −3.97101683524937158435743959909, −2.61647444454060181692975241903, 0.59143899776432484941606518320, 1.57478998957193279536218672530, 3.95371002812051955741869574151, 4.53868569769019495217379944186, 6.09498178415135196849914640881, 8.010201427548553396384716944636, 9.167300936214596725572387606365, 9.928072311483531187127143536873, 11.28902722112039081253993145381, 12.19636285858230916985597513040

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