Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.753 - 0.657i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.15 + 2.08i)3-s + (2.92 + 0.515i)5-s + (0.715 − 0.600i)7-s + (0.293 + 8.99i)9-s + (5.89 − 1.04i)11-s + (−9.53 + 3.47i)13-s + (5.22 + 7.21i)15-s + (6.81 + 3.93i)17-s + (−2.29 − 3.97i)19-s + (2.79 + 0.198i)21-s + (22.9 − 27.3i)23-s + (−15.2 − 5.53i)25-s + (−18.1 + 20.0i)27-s + (3.20 − 8.80i)29-s + (−41.5 − 34.8i)31-s + ⋯
L(s)  = 1  + (0.718 + 0.695i)3-s + (0.584 + 0.103i)5-s + (0.102 − 0.0858i)7-s + (0.0325 + 0.999i)9-s + (0.536 − 0.0945i)11-s + (−0.733 + 0.266i)13-s + (0.348 + 0.480i)15-s + (0.400 + 0.231i)17-s + (−0.120 − 0.208i)19-s + (0.133 + 0.00946i)21-s + (0.996 − 1.18i)23-s + (−0.608 − 0.221i)25-s + (−0.671 + 0.740i)27-s + (0.110 − 0.303i)29-s + (−1.34 − 1.12i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.753 - 0.657i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (77, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.753 - 0.657i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.67721 + 0.629324i\)
\(L(\frac12)\)  \(\approx\)  \(1.67721 + 0.629324i\)
\(L(2)\)   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 \( 1 + (-2.15 - 2.08i)T \)
good5 \( 1 + (-2.92 - 0.515i)T + (23.4 + 8.55i)T^{2} \)
7 \( 1 + (-0.715 + 0.600i)T + (8.50 - 48.2i)T^{2} \)
11 \( 1 + (-5.89 + 1.04i)T + (113. - 41.3i)T^{2} \)
13 \( 1 + (9.53 - 3.47i)T + (129. - 108. i)T^{2} \)
17 \( 1 + (-6.81 - 3.93i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (2.29 + 3.97i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-22.9 + 27.3i)T + (-91.8 - 520. i)T^{2} \)
29 \( 1 + (-3.20 + 8.80i)T + (-644. - 540. i)T^{2} \)
31 \( 1 + (41.5 + 34.8i)T + (166. + 946. i)T^{2} \)
37 \( 1 + (9.21 - 15.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (3.70 + 10.1i)T + (-1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (9.70 + 55.0i)T + (-1.73e3 + 632. i)T^{2} \)
47 \( 1 + (46.8 + 55.8i)T + (-383. + 2.17e3i)T^{2} \)
53 \( 1 - 44.8iT - 2.80e3T^{2} \)
59 \( 1 + (-81.5 - 14.3i)T + (3.27e3 + 1.19e3i)T^{2} \)
61 \( 1 + (-47.6 + 40.0i)T + (646. - 3.66e3i)T^{2} \)
67 \( 1 + (31.9 - 11.6i)T + (3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (-60.7 - 35.0i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-68.4 - 118. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-41.4 - 15.0i)T + (4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (31.4 - 86.3i)T + (-5.27e3 - 4.42e3i)T^{2} \)
89 \( 1 + (86.0 - 49.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (4.81 + 27.3i)T + (-8.84e3 + 3.21e3i)T^{2} \)
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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

−13.82411197793394780363409940962, −12.71600099673135740150597724388, −11.30705131002752665012115815007, −10.18150227966290416041505283950, −9.387163118867926549218511533283, −8.331850862708070378015084009172, −6.94276446572913854949474990525, −5.33727737674622295254198590478, −3.96393552902884883594291045892, −2.31589621078360889915074685292, 1.67043085076627236462164931177, 3.33090701394931209197481012242, 5.32292029026021940799076214858, 6.76086599316342567780034507946, 7.78513464429845160532933099853, 9.075317733262861305461025772672, 9.828732652787052213494322390855, 11.43619587094101236120595217149, 12.53930420886740557502126957382, 13.31820782341652741305205843662

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