Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.0157 + 0.999i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.710 + 1.86i)2-s + (−2.98 − 2.65i)4-s + (−1.35 − 2.34i)5-s + (−10.0 − 5.79i)7-s + (7.09 − 3.70i)8-s + (5.35 − 0.865i)10-s + (−8.54 − 4.93i)11-s + (0.296 + 0.513i)13-s + (17.9 − 14.6i)14-s + (1.87 + 15.8i)16-s + 8.87·17-s − 14.0i·19-s + (−2.18 + 10.6i)20-s + (15.3 − 12.4i)22-s + (−18.2 + 10.5i)23-s + ⋯
L(s)  = 1  + (−0.355 + 0.934i)2-s + (−0.747 − 0.664i)4-s + (−0.271 − 0.469i)5-s + (−1.43 − 0.828i)7-s + (0.886 − 0.462i)8-s + (0.535 − 0.0865i)10-s + (−0.777 − 0.448i)11-s + (0.0227 + 0.0394i)13-s + (1.28 − 1.04i)14-s + (0.117 + 0.993i)16-s + 0.522·17-s − 0.742i·19-s + (−0.109 + 0.531i)20-s + (0.695 − 0.566i)22-s + (−0.794 + 0.458i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0157 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0157 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.0157 + 0.999i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.0157 + 0.999i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.300441 - 0.305213i\)
\(L(\frac12)\)  \(\approx\)  \(0.300441 - 0.305213i\)
\(L(2)\)   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 \( 1 + (0.710 - 1.86i)T \)
3 \( 1 \)
good5 \( 1 + (1.35 + 2.34i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (10.0 + 5.79i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (8.54 + 4.93i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-0.296 - 0.513i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 8.87T + 289T^{2} \)
19 \( 1 + 14.0iT - 361T^{2} \)
23 \( 1 + (18.2 - 10.5i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (10.1 - 17.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (14.3 - 8.27i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 40.6T + 1.36e3T^{2} \)
41 \( 1 + (21.2 + 36.7i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-32.2 - 18.6i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-1.57 - 0.907i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 21.1T + 2.80e3T^{2} \)
59 \( 1 + (-76.6 + 44.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-36.4 + 63.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-38.3 + 22.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 111. iT - 5.04e3T^{2} \)
73 \( 1 + 76.2T + 5.32e3T^{2} \)
79 \( 1 + (-8.30 - 4.79i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-73.6 - 42.5i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 64.7T + 7.92e3T^{2} \)
97 \( 1 + (3.59 - 6.22i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.35308507834820967182704857030, −12.52835266817999078757743796764, −10.70225156803713055254989674283, −9.838806228461882050274509387994, −8.762431080221061891579881428447, −7.56808674998814796905259977031, −6.56799104901549927461298314951, −5.26328382766639237060298772308, −3.68131866756282995835750877444, −0.34330138188566151270547277990, 2.52533217853251119660899958403, 3.69949853456844443332037280636, 5.63437346782765750225975836091, 7.24657426859682535916467727351, 8.539124450150878697522872653346, 9.750200752230068660560944781333, 10.36308570006589550478661892547, 11.73459552508565258001214688868, 12.55492451791512360765950024653, 13.29301332729146811004954709933

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