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

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

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