Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.948 - 0.315i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.31 − 2.69i)3-s + (−5.13 − 0.905i)5-s + (−8.93 + 7.49i)7-s + (−5.51 + 7.10i)9-s + (16.7 − 2.95i)11-s + (−12.2 + 4.44i)13-s + (4.33 + 15.0i)15-s + (−24.5 − 14.1i)17-s + (−13.9 − 24.1i)19-s + (31.9 + 14.1i)21-s + (6.09 − 7.25i)23-s + (2.08 + 0.757i)25-s + (26.4 + 5.48i)27-s + (4.43 − 12.1i)29-s + (11.4 + 9.62i)31-s + ⋯
L(s)  = 1  + (−0.439 − 0.898i)3-s + (−1.02 − 0.181i)5-s + (−1.27 + 1.07i)7-s + (−0.613 + 0.789i)9-s + (1.52 − 0.268i)11-s + (−0.939 + 0.341i)13-s + (0.289 + 1.00i)15-s + (−1.44 − 0.834i)17-s + (−0.733 − 1.27i)19-s + (1.52 + 0.675i)21-s + (0.264 − 0.315i)23-s + (0.0832 + 0.0303i)25-s + (0.979 + 0.203i)27-s + (0.153 − 0.420i)29-s + (0.370 + 0.310i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.315i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.948 - 0.315i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (77, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.948 - 0.315i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0160103 + 0.0988713i\)
\(L(\frac12)\)  \(\approx\)  \(0.0160103 + 0.0988713i\)
\(L(2)\)   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 \( 1 + (1.31 + 2.69i)T \)
good5 \( 1 + (5.13 + 0.905i)T + (23.4 + 8.55i)T^{2} \)
7 \( 1 + (8.93 - 7.49i)T + (8.50 - 48.2i)T^{2} \)
11 \( 1 + (-16.7 + 2.95i)T + (113. - 41.3i)T^{2} \)
13 \( 1 + (12.2 - 4.44i)T + (129. - 108. i)T^{2} \)
17 \( 1 + (24.5 + 14.1i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (13.9 + 24.1i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-6.09 + 7.25i)T + (-91.8 - 520. i)T^{2} \)
29 \( 1 + (-4.43 + 12.1i)T + (-644. - 540. i)T^{2} \)
31 \( 1 + (-11.4 - 9.62i)T + (166. + 946. i)T^{2} \)
37 \( 1 + (19.3 - 33.4i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (0.663 + 1.82i)T + (-1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (-9.15 - 51.8i)T + (-1.73e3 + 632. i)T^{2} \)
47 \( 1 + (-1.66 - 1.98i)T + (-383. + 2.17e3i)T^{2} \)
53 \( 1 + 14.7iT - 2.80e3T^{2} \)
59 \( 1 + (16.7 + 2.96i)T + (3.27e3 + 1.19e3i)T^{2} \)
61 \( 1 + (17.6 - 14.7i)T + (646. - 3.66e3i)T^{2} \)
67 \( 1 + (-50.8 + 18.5i)T + (3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (73.9 + 42.6i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (1.73 + 3.00i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (15.2 + 5.54i)T + (4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (31.4 - 86.4i)T + (-5.27e3 - 4.42e3i)T^{2} \)
89 \( 1 + (55.7 - 32.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-7.70 - 43.6i)T + (-8.84e3 + 3.21e3i)T^{2} \)
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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

−12.64820550777744508820891851161, −11.90392116878124695375609820434, −11.31244003986529718448372029357, −9.369739446233303818816366191501, −8.579764519448023726838416149802, −6.93656546849231920117163117731, −6.40906918804332433233310302161, −4.60302697710901643560836742234, −2.67556117263918848827297239937, −0.07407348898484596878082526836, 3.69562651164830663089270973460, 4.20591806108513592907327800809, 6.26379968916086650820450028935, 7.19427906310310268166648756109, 8.860512385007333451944070050512, 9.964239121487113779890860695463, 10.76392872421045471363809718989, 11.89346382956203107348304464829, 12.75709577191358175879066966230, 14.27945883287020820645956411373

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