Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.851 + 0.525i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.86 − 2.35i)3-s + (7.10 + 1.25i)5-s + (−3.36 + 2.82i)7-s + (−2.07 − 8.75i)9-s + (−6.85 + 1.20i)11-s + (19.7 − 7.18i)13-s + (16.1 − 14.3i)15-s + (−21.7 − 12.5i)17-s + (11.6 + 20.1i)19-s + (0.382 + 13.1i)21-s + (−17.3 + 20.6i)23-s + (25.4 + 9.24i)25-s + (−24.4 − 11.4i)27-s + (−3.00 + 8.26i)29-s + (−23.8 − 19.9i)31-s + ⋯
L(s)  = 1  + (0.620 − 0.784i)3-s + (1.42 + 0.250i)5-s + (−0.480 + 0.403i)7-s + (−0.230 − 0.973i)9-s + (−0.623 + 0.109i)11-s + (1.51 − 0.552i)13-s + (1.07 − 0.959i)15-s + (−1.28 − 0.739i)17-s + (0.611 + 1.05i)19-s + (0.0182 + 0.627i)21-s + (−0.754 + 0.898i)23-s + (1.01 + 0.369i)25-s + (−0.906 − 0.422i)27-s + (−0.103 + 0.285i)29-s + (−0.767 − 0.644i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.851 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.851 + 0.525i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (77, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.851 + 0.525i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.77098 - 0.502398i\)
\(L(\frac12)\)  \(\approx\)  \(1.77098 - 0.502398i\)
\(L(2)\)   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 \( 1 + (-1.86 + 2.35i)T \)
good5 \( 1 + (-7.10 - 1.25i)T + (23.4 + 8.55i)T^{2} \)
7 \( 1 + (3.36 - 2.82i)T + (8.50 - 48.2i)T^{2} \)
11 \( 1 + (6.85 - 1.20i)T + (113. - 41.3i)T^{2} \)
13 \( 1 + (-19.7 + 7.18i)T + (129. - 108. i)T^{2} \)
17 \( 1 + (21.7 + 12.5i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-11.6 - 20.1i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (17.3 - 20.6i)T + (-91.8 - 520. i)T^{2} \)
29 \( 1 + (3.00 - 8.26i)T + (-644. - 540. i)T^{2} \)
31 \( 1 + (23.8 + 19.9i)T + (166. + 946. i)T^{2} \)
37 \( 1 + (13.9 - 24.1i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-10.2 - 28.2i)T + (-1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (-8.18 - 46.4i)T + (-1.73e3 + 632. i)T^{2} \)
47 \( 1 + (33.0 + 39.3i)T + (-383. + 2.17e3i)T^{2} \)
53 \( 1 - 37.5iT - 2.80e3T^{2} \)
59 \( 1 + (19.7 + 3.47i)T + (3.27e3 + 1.19e3i)T^{2} \)
61 \( 1 + (-74.5 + 62.5i)T + (646. - 3.66e3i)T^{2} \)
67 \( 1 + (-70.1 + 25.5i)T + (3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (51.3 + 29.6i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (37.3 + 64.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-90.4 - 32.9i)T + (4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (-42.2 + 115. i)T + (-5.27e3 - 4.42e3i)T^{2} \)
89 \( 1 + (17.3 - 9.99i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (2.12 + 12.0i)T + (-8.84e3 + 3.21e3i)T^{2} \)
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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

−13.43708710376454377989010502590, −12.76833334281390555116549457342, −11.29889255520632643026037760507, −9.928178648211011309739790042045, −9.114633049008948616310031831385, −7.903233766992442535239876385048, −6.44161028259255629896840733615, −5.71037684493839329410052530447, −3.19801101041057420188914607720, −1.83006607846911033783963968937, 2.21356536669881047307042176458, 3.93467906309787927697400465744, 5.43247357415304508424239801486, 6.65780802098325294297557176224, 8.546083139579452949813038905569, 9.225340067501681078945732819853, 10.30798260904251280909862129697, 11.01653044109648901439738841500, 13.10161310924040972187967115090, 13.49558684065286353881548804791

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