Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.814 - 0.580i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.88 − 2.10i)2-s + (−4.43 + 2.71i)3-s + (−0.902 + 7.94i)4-s + (5.32 − 14.6i)5-s + (14.0 + 4.24i)6-s + (6.67 + 1.17i)7-s + (18.4 − 13.0i)8-s + (12.3 − 24.0i)9-s + (−40.9 + 16.3i)10-s + (−32.8 + 11.9i)11-s + (−17.5 − 37.6i)12-s + (−58.0 + 48.7i)13-s + (−10.0 − 16.3i)14-s + (16.0 + 79.3i)15-s + (−62.3 − 14.3i)16-s + (−26.5 + 15.2i)17-s + ⋯
L(s)  = 1  + (−0.666 − 0.745i)2-s + (−0.853 + 0.521i)3-s + (−0.112 + 0.993i)4-s + (0.476 − 1.30i)5-s + (0.957 + 0.288i)6-s + (0.360 + 0.0635i)7-s + (0.816 − 0.577i)8-s + (0.455 − 0.890i)9-s + (−1.29 + 0.516i)10-s + (−0.900 + 0.327i)11-s + (−0.422 − 0.906i)12-s + (−1.23 + 1.04i)13-s + (−0.192 − 0.311i)14-s + (0.276 + 1.36i)15-s + (−0.974 − 0.224i)16-s + (−0.378 + 0.218i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.814 - 0.580i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.814 - 0.580i)\)
\(L(2)\)  \(\approx\)  \(0.0195386 + 0.0610780i\)
\(L(\frac12)\)  \(\approx\)  \(0.0195386 + 0.0610780i\)
\(L(\frac{5}{2})\)   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 \( 1 + (1.88 + 2.10i)T \)
3 \( 1 + (4.43 - 2.71i)T \)
good5 \( 1 + (-5.32 + 14.6i)T + (-95.7 - 80.3i)T^{2} \)
7 \( 1 + (-6.67 - 1.17i)T + (322. + 117. i)T^{2} \)
11 \( 1 + (32.8 - 11.9i)T + (1.01e3 - 855. i)T^{2} \)
13 \( 1 + (58.0 - 48.7i)T + (381. - 2.16e3i)T^{2} \)
17 \( 1 + (26.5 - 15.2i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (47.8 + 27.6i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (16.2 + 91.9i)T + (-1.14e4 + 4.16e3i)T^{2} \)
29 \( 1 + (146. - 174. i)T + (-4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (134. - 23.6i)T + (2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (109. + 189. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-42.2 - 50.2i)T + (-1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (-37.6 - 103. i)T + (-6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (1.93 - 10.9i)T + (-9.75e4 - 3.55e4i)T^{2} \)
53 \( 1 + 586. iT - 1.48e5T^{2} \)
59 \( 1 + (-380. - 138. i)T + (1.57e5 + 1.32e5i)T^{2} \)
61 \( 1 + (122. - 695. i)T + (-2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (-518. - 617. i)T + (-5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (526. + 911. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-335. + 581. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (715. - 852. i)T + (-8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (539. + 452. i)T + (9.92e4 + 5.63e5i)T^{2} \)
89 \( 1 + (-731. - 422. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-930. + 338. i)T + (6.99e5 - 5.86e5i)T^{2} \)
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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

−12.53412358651778595795238603822, −11.46665543858666905398618214701, −10.43010966611911704496107375487, −9.457090453474381484993699003875, −8.682358539980900092511834554028, −7.10136388566883881584268567081, −5.18636421401821871815613087944, −4.38360884428628799143546408506, −1.91192421603864829817263100612, −0.04573571756737732110792078637, 2.25328268417482511565483926110, 5.18934007961421201879249057952, 6.08713990629440379408643494299, 7.25923165936804029479032571258, 7.904636125631270529559923481752, 9.857998334472726334291966596513, 10.57497022550822413994295246204, 11.35026029250082266311875960489, 12.96898355600520055232352222158, 13.99125498050781951124238526599

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