Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.999 + 0.0149i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.14 + 0.728i)3-s + (−5.18 − 4.34i)5-s + (16.8 + 6.12i)7-s + (25.9 + 7.49i)9-s + (45.5 − 38.2i)11-s + (−5.02 + 28.5i)13-s + (−23.4 − 26.1i)15-s + (−14.6 + 25.4i)17-s + (13.7 + 23.8i)19-s + (82.0 + 43.7i)21-s + (−0.946 + 0.344i)23-s + (−13.7 − 78.0i)25-s + (127. + 57.4i)27-s + (−15.4 − 87.8i)29-s + (−11.6 + 4.23i)31-s + ⋯
L(s)  = 1  + (0.990 + 0.140i)3-s + (−0.463 − 0.388i)5-s + (0.908 + 0.330i)7-s + (0.960 + 0.277i)9-s + (1.24 − 1.04i)11-s + (−0.107 + 0.608i)13-s + (−0.404 − 0.450i)15-s + (−0.209 + 0.362i)17-s + (0.166 + 0.288i)19-s + (0.853 + 0.454i)21-s + (−0.00857 + 0.00312i)23-s + (−0.110 − 0.624i)25-s + (0.912 + 0.409i)27-s + (−0.0991 − 0.562i)29-s + (−0.0674 + 0.0245i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0149i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.999 + 0.0149i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 0.999 + 0.0149i)\)
\(L(2)\)  \(\approx\)  \(2.27321 - 0.0169982i\)
\(L(\frac12)\)  \(\approx\)  \(2.27321 - 0.0169982i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-5.14 - 0.728i)T \)
good5 \( 1 + (5.18 + 4.34i)T + (21.7 + 123. i)T^{2} \)
7 \( 1 + (-16.8 - 6.12i)T + (262. + 220. i)T^{2} \)
11 \( 1 + (-45.5 + 38.2i)T + (231. - 1.31e3i)T^{2} \)
13 \( 1 + (5.02 - 28.5i)T + (-2.06e3 - 751. i)T^{2} \)
17 \( 1 + (14.6 - 25.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-13.7 - 23.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (0.946 - 0.344i)T + (9.32e3 - 7.82e3i)T^{2} \)
29 \( 1 + (15.4 + 87.8i)T + (-2.29e4 + 8.34e3i)T^{2} \)
31 \( 1 + (11.6 - 4.23i)T + (2.28e4 - 1.91e4i)T^{2} \)
37 \( 1 + (186. - 323. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-38.3 + 217. i)T + (-6.47e4 - 2.35e4i)T^{2} \)
43 \( 1 + (415. - 348. i)T + (1.38e4 - 7.82e4i)T^{2} \)
47 \( 1 + (575. + 209. i)T + (7.95e4 + 6.67e4i)T^{2} \)
53 \( 1 + 51.1T + 1.48e5T^{2} \)
59 \( 1 + (326. + 274. i)T + (3.56e4 + 2.02e5i)T^{2} \)
61 \( 1 + (-488. - 177. i)T + (1.73e5 + 1.45e5i)T^{2} \)
67 \( 1 + (-38.0 + 215. i)T + (-2.82e5 - 1.02e5i)T^{2} \)
71 \( 1 + (99.9 - 173. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (398. + 690. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-227. - 1.28e3i)T + (-4.63e5 + 1.68e5i)T^{2} \)
83 \( 1 + (11.1 + 63.3i)T + (-5.37e5 + 1.95e5i)T^{2} \)
89 \( 1 + (-466. - 807. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-915. + 768. i)T + (1.58e5 - 8.98e5i)T^{2} \)
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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

−13.46859675420045193574667504863, −12.06335214759520000804755464611, −11.30791498051649860902389580775, −9.779534486024294348845210665556, −8.604265574384475270480596318485, −8.166508575271750368212110739664, −6.57045042555440384053821516003, −4.72698680524994906629592937466, −3.56261506416434569548452429529, −1.62985676844119894260007125159, 1.68455800644537010332777776673, 3.48110656388023908465334317076, 4.71936512001933650687696752196, 6.92168515817976191145585418547, 7.63200667923822270716349599164, 8.823249786592940668644211210047, 9.885565870777904464605395674699, 11.17252337432906867984864923239, 12.21248303195687968342150200960, 13.35869253981461362242943863787

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