Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.0834 - 0.996i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.82 − 4.16i)2-s + (−2.67 + 31.8i)4-s + 46.0i·5-s − 134. i·7-s + (142. − 110. i)8-s + (191. − 176. i)10-s + 471.·11-s − 987.·13-s + (−561. + 516. i)14-s + (−1.00e3 − 170. i)16-s + 1.46e3i·17-s − 308. i·19-s + (−1.46e3 − 122. i)20-s + (−1.80e3 − 1.96e3i)22-s − 2.11e3·23-s + ⋯
L(s)  = 1  + (−0.676 − 0.736i)2-s + (−0.0834 + 0.996i)4-s + 0.823i·5-s − 1.03i·7-s + (0.789 − 0.613i)8-s + (0.606 − 0.557i)10-s + 1.17·11-s − 1.61·13-s + (−0.765 + 0.703i)14-s + (−0.986 − 0.166i)16-s + 1.23i·17-s − 0.195i·19-s + (−0.820 − 0.0687i)20-s + (−0.795 − 0.864i)22-s − 0.835·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0834 - 0.996i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0834 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.0834 - 0.996i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ 0.0834 - 0.996i)\)
\(L(3)\)  \(\approx\)  \(0.495754 + 0.455972i\)
\(L(\frac12)\)  \(\approx\)  \(0.495754 + 0.455972i\)
\(L(\frac{7}{2})\)   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 \( 1 + (3.82 + 4.16i)T \)
3 \( 1 \)
good5 \( 1 - 46.0iT - 3.12e3T^{2} \)
7 \( 1 + 134. iT - 1.68e4T^{2} \)
11 \( 1 - 471.T + 1.61e5T^{2} \)
13 \( 1 + 987.T + 3.71e5T^{2} \)
17 \( 1 - 1.46e3iT - 1.41e6T^{2} \)
19 \( 1 + 308. iT - 2.47e6T^{2} \)
23 \( 1 + 2.11e3T + 6.43e6T^{2} \)
29 \( 1 - 5.19e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.36e3iT - 2.86e7T^{2} \)
37 \( 1 + 8.74e3T + 6.93e7T^{2} \)
41 \( 1 - 1.39e3iT - 1.15e8T^{2} \)
43 \( 1 - 9.41e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.11e4T + 2.29e8T^{2} \)
53 \( 1 - 2.84e4iT - 4.18e8T^{2} \)
59 \( 1 - 1.50e4T + 7.14e8T^{2} \)
61 \( 1 + 3.78e4T + 8.44e8T^{2} \)
67 \( 1 - 6.52e4iT - 1.35e9T^{2} \)
71 \( 1 - 9.86e3T + 1.80e9T^{2} \)
73 \( 1 + 5.48e4T + 2.07e9T^{2} \)
79 \( 1 + 9.16e4iT - 3.07e9T^{2} \)
83 \( 1 + 4.51e4T + 3.93e9T^{2} \)
89 \( 1 + 7.31e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.14e4T + 8.58e9T^{2} \)
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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

−12.70438634573902899378586166282, −11.83127439058494580629679078170, −10.62730831725798555303721152834, −10.12768669321800024944653840827, −8.864263486633500337200557610811, −7.45678807161504904053631694614, −6.72977466683333338622401846004, −4.36512384361181178977590773678, −3.16317148869674216152927259022, −1.50439025118279309023963818542, 0.32408973754381589247151495447, 2.09374092474360616583270230193, 4.65996312417106047607612008764, 5.67641082798800930494886885646, 6.99461391839535744568351476989, 8.228793444583218650165621230237, 9.269570883185514233505547396204, 9.788061235114596076390892351666, 11.63038989392557281790676112257, 12.29208219836791413133954982168

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