Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.744 + 0.668i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (34.8 − 20.1i)5-s + (−7.38 + 12.7i)7-s + (70.7 + 40.8i)11-s + (−139. − 240. i)13-s − 10.8i·17-s + 532.·19-s + (702. − 405. i)23-s + (498. − 862. i)25-s + (257. + 148. i)29-s + (−97.5 − 168. i)31-s + 594. i·35-s − 2.09e3·37-s + (1.35e3 − 784. i)41-s + (46.0 − 79.8i)43-s + (−1.84e3 − 1.06e3i)47-s + ⋯
L(s)  = 1  + (1.39 − 0.805i)5-s + (−0.150 + 0.261i)7-s + (0.584 + 0.337i)11-s + (−0.822 − 1.42i)13-s − 0.0376i·17-s + 1.47·19-s + (1.32 − 0.766i)23-s + (0.797 − 1.38i)25-s + (0.306 + 0.176i)29-s + (−0.101 − 0.175i)31-s + 0.485i·35-s − 1.53·37-s + (0.808 − 0.466i)41-s + (0.0249 − 0.0431i)43-s + (−0.837 − 0.483i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 + 0.668i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.744 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.744 + 0.668i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (89, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.744 + 0.668i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(2.02961 - 0.777429i\)
\(L(\frac12)\)  \(\approx\)  \(2.02961 - 0.777429i\)
\(L(3)\)   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 \( 1 \)
good5 \( 1 + (-34.8 + 20.1i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (7.38 - 12.7i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-70.7 - 40.8i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (139. + 240. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 10.8iT - 8.35e4T^{2} \)
19 \( 1 - 532.T + 1.30e5T^{2} \)
23 \( 1 + (-702. + 405. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-257. - 148. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (97.5 + 168. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 2.09e3T + 1.87e6T^{2} \)
41 \( 1 + (-1.35e3 + 784. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-46.0 + 79.8i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (1.84e3 + 1.06e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 2.57e3iT - 7.89e6T^{2} \)
59 \( 1 + (1.34e3 - 778. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.68e3 - 4.65e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (457. + 792. i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 8.21e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.43e3T + 2.83e7T^{2} \)
79 \( 1 + (2.31e3 - 4.01e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-5.19e3 - 2.99e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 8.43e3iT - 6.27e7T^{2} \)
97 \( 1 + (-3.01e3 + 5.22e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−12.83922021574040383008569387021, −12.14285085109522343382261966173, −10.48037582134467512731872828812, −9.597577096673841779747433817385, −8.788850227551588733061765017548, −7.23085517260884146574615903176, −5.75454191300585092096786178174, −4.96462948044494793577214220625, −2.77971099993704726223552790299, −1.11538410422801554956599198626, 1.66508312469006032719636975083, 3.19587413054919385278757209766, 5.10856116051280890647854153359, 6.44291075451527455609540896029, 7.22608557294268337872455975576, 9.209981095276986760584231411525, 9.735826407494586052173151780406, 10.94687011375246620993347218001, 11.96732071970606954235613470983, 13.46710480983808918420945582272

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