Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.748 - 0.662i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.17 + 2.57i)2-s + (4.92 + 1.66i)3-s + (−5.25 − 6.03i)4-s + (3.67 − 10.0i)5-s + (−10.0 + 10.7i)6-s + (6.04 + 1.06i)7-s + (21.6 − 6.45i)8-s + (21.4 + 16.4i)9-s + (21.6 + 21.2i)10-s + (31.0 − 11.3i)11-s + (−15.7 − 38.4i)12-s + (33.3 − 27.9i)13-s + (−9.82 + 14.3i)14-s + (34.9 − 43.5i)15-s + (−8.78 + 63.3i)16-s + (−97.6 + 56.3i)17-s + ⋯
L(s)  = 1  + (−0.414 + 0.910i)2-s + (0.947 + 0.321i)3-s + (−0.656 − 0.754i)4-s + (0.328 − 0.902i)5-s + (−0.684 + 0.728i)6-s + (0.326 + 0.0575i)7-s + (0.958 − 0.285i)8-s + (0.793 + 0.608i)9-s + (0.685 + 0.672i)10-s + (0.852 − 0.310i)11-s + (−0.379 − 0.925i)12-s + (0.710 − 0.596i)13-s + (−0.187 + 0.273i)14-s + (0.600 − 0.748i)15-s + (−0.137 + 0.990i)16-s + (−1.39 + 0.804i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 - 0.662i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.748 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.748 - 0.662i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 0.748 - 0.662i)\)
\(L(2)\)  \(\approx\)  \(1.75220 + 0.664229i\)
\(L(\frac12)\)  \(\approx\)  \(1.75220 + 0.664229i\)
\(L(\frac{5}{2})\)   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 \( 1 + (1.17 - 2.57i)T \)
3 \( 1 + (-4.92 - 1.66i)T \)
good5 \( 1 + (-3.67 + 10.0i)T + (-95.7 - 80.3i)T^{2} \)
7 \( 1 + (-6.04 - 1.06i)T + (322. + 117. i)T^{2} \)
11 \( 1 + (-31.0 + 11.3i)T + (1.01e3 - 855. i)T^{2} \)
13 \( 1 + (-33.3 + 27.9i)T + (381. - 2.16e3i)T^{2} \)
17 \( 1 + (97.6 - 56.3i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-87.7 - 50.6i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (28.7 + 163. i)T + (-1.14e4 + 4.16e3i)T^{2} \)
29 \( 1 + (127. - 152. i)T + (-4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (-66.6 + 11.7i)T + (2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (-98.0 - 169. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (133. + 158. i)T + (-1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (98.0 + 269. i)T + (-6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (-10.0 + 56.8i)T + (-9.75e4 - 3.55e4i)T^{2} \)
53 \( 1 + 127. iT - 1.48e5T^{2} \)
59 \( 1 + (190. + 69.3i)T + (1.57e5 + 1.32e5i)T^{2} \)
61 \( 1 + (-64.6 + 366. i)T + (-2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (-310. - 370. i)T + (-5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (-18.1 - 31.4i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (295. - 511. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (697. - 830. i)T + (-8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (-747. - 626. i)T + (9.92e4 + 5.63e5i)T^{2} \)
89 \( 1 + (646. + 373. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (842. - 306. i)T + (6.99e5 - 5.86e5i)T^{2} \)
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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

−13.60628992466094189084803707232, −12.76064397618262331684582069913, −10.87037115197681988731282657990, −9.707134028385995449197309828697, −8.659923320141107533830445975237, −8.331647215654841731577533432177, −6.71371786484140951613478497633, −5.26966732851268292015713407754, −4.00396322512160130555928062820, −1.44675875685670177614840346306, 1.62940436786428581949230624070, 2.93802461697922188009398707232, 4.25460219682639276843596686158, 6.71479480087783594640339208420, 7.72431665804494037097306072302, 9.138126578582327590481060555542, 9.626504584049021435612306362357, 11.13039452664714492888391923931, 11.77669190077345115067539249232, 13.40714063253909392622299242052

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