Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.918 + 0.394i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.05 − 1.18i)5-s + (4.05 − 7.02i)7-s + (17.6 + 10.1i)11-s + (−3.05 − 5.29i)13-s − 17.9i·17-s + 9.11·19-s + (−29.0 + 16.7i)23-s + (−9.67 + 16.7i)25-s + (−14.4 − 8.31i)29-s + (11.1 + 19.3i)31-s − 19.2i·35-s − 50.4·37-s + (−29.9 + 17.3i)41-s + (−11.5 + 19.9i)43-s + (33.1 + 19.1i)47-s + ⋯
L(s)  = 1  + (0.411 − 0.237i)5-s + (0.579 − 1.00i)7-s + (1.60 + 0.924i)11-s + (−0.235 − 0.407i)13-s − 1.05i·17-s + 0.479·19-s + (−1.26 + 0.729i)23-s + (−0.387 + 0.670i)25-s + (−0.496 − 0.286i)29-s + (0.360 + 0.624i)31-s − 0.551i·35-s − 1.36·37-s + (−0.730 + 0.421i)41-s + (−0.267 + 0.463i)43-s + (0.705 + 0.407i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.918 + 0.394i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (89, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.918 + 0.394i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.50849 - 0.309942i\)
\(L(\frac12)\)  \(\approx\)  \(1.50849 - 0.309942i\)
\(L(2)\)   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 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-2.05 + 1.18i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-4.05 + 7.02i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-17.6 - 10.1i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (3.05 + 5.29i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 17.9iT - 289T^{2} \)
19 \( 1 - 9.11T + 361T^{2} \)
23 \( 1 + (29.0 - 16.7i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (14.4 + 8.31i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-11.1 - 19.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 50.4T + 1.36e3T^{2} \)
41 \( 1 + (29.9 - 17.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (11.5 - 19.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-33.1 - 19.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 19.0iT - 2.80e3T^{2} \)
59 \( 1 + (-2.96 + 1.71i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-23.1 + 40.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-3.14 - 5.45i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 35.9iT - 5.04e3T^{2} \)
73 \( 1 - 47.3T + 5.32e3T^{2} \)
79 \( 1 + (-42.2 + 73.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-33.1 - 19.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 143. iT - 7.92e3T^{2} \)
97 \( 1 + (40.3 - 69.9i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.67566831544782780617979981701, −12.23360305825530446884858886265, −11.42648792657466529221595355620, −10.03733274293867804816700674026, −9.303799228602202580264676005988, −7.74298474067207329194164358022, −6.78668673138639759254996227644, −5.15105172834436776771352073313, −3.86893486543229514033237225567, −1.51334409011988312388596239121, 1.95745800377030199383279169492, 3.90061714212670349806568205242, 5.65726420548311825099845956703, 6.55764872325180512179548165730, 8.305428035199002488750902065644, 9.088044384700959978361099416781, 10.35544830476288545429801970998, 11.66695204472396881505891769605, 12.19024726569963303498998603180, 13.81913363892772358719792428168

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