Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.850 + 0.526i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.72 + 0.773i)2-s + (6.80 − 4.21i)4-s − 20.8i·5-s + 13.9i·7-s + (−15.2 + 16.7i)8-s + (16.1 + 56.7i)10-s − 34.5·11-s − 31.3·13-s + (−10.7 − 37.8i)14-s + (28.5 − 57.2i)16-s + 34.4i·17-s − 120. i·19-s + (−87.7 − 141. i)20-s + (93.8 − 26.7i)22-s − 137.·23-s + ⋯
L(s)  = 1  + (−0.961 + 0.273i)2-s + (0.850 − 0.526i)4-s − 1.86i·5-s + 0.750i·7-s + (−0.673 + 0.738i)8-s + (0.510 + 1.79i)10-s − 0.945·11-s − 0.668·13-s + (−0.205 − 0.722i)14-s + (0.445 − 0.895i)16-s + 0.491i·17-s − 1.45i·19-s + (−0.981 − 1.58i)20-s + (0.909 − 0.258i)22-s − 1.24·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.850 + 0.526i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.850 + 0.526i)\)
\(L(2)\)  \(\approx\)  \(0.120282 - 0.422795i\)
\(L(\frac12)\)  \(\approx\)  \(0.120282 - 0.422795i\)
\(L(\frac{5}{2})\)   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 \( 1 + (2.72 - 0.773i)T \)
3 \( 1 \)
good5 \( 1 + 20.8iT - 125T^{2} \)
7 \( 1 - 13.9iT - 343T^{2} \)
11 \( 1 + 34.5T + 1.33e3T^{2} \)
13 \( 1 + 31.3T + 2.19e3T^{2} \)
17 \( 1 - 34.4iT - 4.91e3T^{2} \)
19 \( 1 + 120. iT - 6.85e3T^{2} \)
23 \( 1 + 137.T + 1.21e4T^{2} \)
29 \( 1 + 93.1iT - 2.43e4T^{2} \)
31 \( 1 - 111. iT - 2.97e4T^{2} \)
37 \( 1 + 146.T + 5.06e4T^{2} \)
41 \( 1 + 8.44iT - 6.89e4T^{2} \)
43 \( 1 + 427. iT - 7.95e4T^{2} \)
47 \( 1 - 318.T + 1.03e5T^{2} \)
53 \( 1 - 291. iT - 1.48e5T^{2} \)
59 \( 1 + 364.T + 2.05e5T^{2} \)
61 \( 1 + 289.T + 2.26e5T^{2} \)
67 \( 1 + 305. iT - 3.00e5T^{2} \)
71 \( 1 - 102.T + 3.57e5T^{2} \)
73 \( 1 - 442.T + 3.89e5T^{2} \)
79 \( 1 + 245. iT - 4.93e5T^{2} \)
83 \( 1 - 478.T + 5.71e5T^{2} \)
89 \( 1 + 1.41e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.15e3T + 9.12e5T^{2} \)
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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

−12.51273512383937757227258966451, −11.89255195765867715148052979368, −10.38131376855891137862894583246, −9.217803647424019148046820922053, −8.606778924466525929943087186890, −7.60522181874710645815746440140, −5.80694550585401314966782390789, −4.87981700063386842384458082735, −2.12991481930781626514433842052, −0.30432762571761501793149754270, 2.33707485222832699916042660376, 3.57949107565351646971696693997, 6.15473072863399169093223001726, 7.32405486742889931902126593435, 7.87752696595028383643677735288, 9.862087619822094309370609722745, 10.34550443990491264556600663211, 11.16337224487141225208746933801, 12.27917939197709249807105203494, 13.81695595146035647867261865578

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