Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.749 + 0.661i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.16 + 0.546i)3-s + (5.21 − 1.89i)5-s + (1.58 + 9.00i)7-s + (26.4 − 5.64i)9-s + (−67.4 − 24.5i)11-s + (−63.1 − 53.0i)13-s + (−25.9 + 12.6i)15-s + (21.0 − 36.4i)17-s + (−22.5 − 39.0i)19-s + (−13.1 − 45.6i)21-s + (12.1 − 68.7i)23-s + (−72.1 + 60.5i)25-s + (−133. + 43.6i)27-s + (142. − 119. i)29-s + (−46.3 + 262. i)31-s + ⋯
L(s)  = 1  + (−0.994 + 0.105i)3-s + (0.466 − 0.169i)5-s + (0.0857 + 0.486i)7-s + (0.977 − 0.209i)9-s + (−1.85 − 0.673i)11-s + (−1.34 − 1.13i)13-s + (−0.446 + 0.217i)15-s + (0.299 − 0.519i)17-s + (−0.271 − 0.471i)19-s + (−0.136 − 0.474i)21-s + (0.109 − 0.623i)23-s + (−0.577 + 0.484i)25-s + (−0.950 + 0.310i)27-s + (0.913 − 0.766i)29-s + (−0.268 + 1.52i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.749 + 0.661i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (49, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.749 + 0.661i)\)
\(L(2)\)  \(\approx\)  \(0.150393 - 0.397588i\)
\(L(\frac12)\)  \(\approx\)  \(0.150393 - 0.397588i\)
\(L(\frac{5}{2})\)   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 \( 1 \)
3 \( 1 + (5.16 - 0.546i)T \)
good5 \( 1 + (-5.21 + 1.89i)T + (95.7 - 80.3i)T^{2} \)
7 \( 1 + (-1.58 - 9.00i)T + (-322. + 117. i)T^{2} \)
11 \( 1 + (67.4 + 24.5i)T + (1.01e3 + 855. i)T^{2} \)
13 \( 1 + (63.1 + 53.0i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (-21.0 + 36.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (22.5 + 39.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-12.1 + 68.7i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (-142. + 119. i)T + (4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (46.3 - 262. i)T + (-2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (40.4 - 70.0i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (216. + 181. i)T + (1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (249. + 90.9i)T + (6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (10.5 + 59.8i)T + (-9.75e4 + 3.55e4i)T^{2} \)
53 \( 1 - 206.T + 1.48e5T^{2} \)
59 \( 1 + (-558. + 203. i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (-124. - 706. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (-23.6 - 19.8i)T + (5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (-155. + 269. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (263. + 457. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (500. - 420. i)T + (8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (883. - 741. i)T + (9.92e4 - 5.63e5i)T^{2} \)
89 \( 1 + (-240. - 417. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (858. + 312. i)T + (6.99e5 + 5.86e5i)T^{2} \)
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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

−12.72903787986577082673610373721, −11.85994670481888743905883491906, −10.51979840361603379638944018578, −10.02042180938651256774411487687, −8.386049634024690926232484495888, −7.11744069903886496203703948646, −5.49023287230430565373365026912, −5.11715437775329451297434125938, −2.67364281647306529076441056169, −0.24624619207289681444294600953, 2.08794427169218524383836782808, 4.49896453233113016287626100810, 5.53967783937979136940461276284, 6.91256674435146056007962899163, 7.82619082226860449591613412192, 9.895487135338227985551703588821, 10.27539947855210021707515701048, 11.55124059290266920256655440966, 12.57295666383077670650599147782, 13.40243344605391017008047105834

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