Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.252 + 0.967i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.92 + 0.656i)3-s + (0.740 − 2.03i)5-s + (1.08 − 6.13i)7-s + (8.13 − 3.84i)9-s + (−5.10 − 14.0i)11-s + (9.95 − 8.35i)13-s + (−0.831 + 6.44i)15-s + (−3.36 + 1.94i)17-s + (6.39 − 11.0i)19-s + (0.863 + 18.6i)21-s + (−35.3 + 6.23i)23-s + (15.5 + 13.0i)25-s + (−21.2 + 16.6i)27-s + (7.13 − 8.49i)29-s + (6.25 + 35.4i)31-s + ⋯
L(s)  = 1  + (−0.975 + 0.218i)3-s + (0.148 − 0.407i)5-s + (0.154 − 0.877i)7-s + (0.904 − 0.427i)9-s + (−0.464 − 1.27i)11-s + (0.766 − 0.642i)13-s + (−0.0554 + 0.429i)15-s + (−0.197 + 0.114i)17-s + (0.336 − 0.583i)19-s + (0.0411 + 0.889i)21-s + (−1.53 + 0.271i)23-s + (0.622 + 0.522i)25-s + (−0.788 + 0.614i)27-s + (0.245 − 0.293i)29-s + (0.201 + 1.14i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.252 + 0.967i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (65, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.252 + 0.967i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.738790 - 0.570892i\)
\(L(\frac12)\)  \(\approx\)  \(0.738790 - 0.570892i\)
\(L(2)\)   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 \( 1 + (2.92 - 0.656i)T \)
good5 \( 1 + (-0.740 + 2.03i)T + (-19.1 - 16.0i)T^{2} \)
7 \( 1 + (-1.08 + 6.13i)T + (-46.0 - 16.7i)T^{2} \)
11 \( 1 + (5.10 + 14.0i)T + (-92.6 + 77.7i)T^{2} \)
13 \( 1 + (-9.95 + 8.35i)T + (29.3 - 166. i)T^{2} \)
17 \( 1 + (3.36 - 1.94i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.39 + 11.0i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (35.3 - 6.23i)T + (497. - 180. i)T^{2} \)
29 \( 1 + (-7.13 + 8.49i)T + (-146. - 828. i)T^{2} \)
31 \( 1 + (-6.25 - 35.4i)T + (-903. + 328. i)T^{2} \)
37 \( 1 + (16.3 + 28.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (14.5 + 17.3i)T + (-291. + 1.65e3i)T^{2} \)
43 \( 1 + (-58.2 + 21.2i)T + (1.41e3 - 1.18e3i)T^{2} \)
47 \( 1 + (3.36 + 0.592i)T + (2.07e3 + 755. i)T^{2} \)
53 \( 1 - 79.3iT - 2.80e3T^{2} \)
59 \( 1 + (-7.36 + 20.2i)T + (-2.66e3 - 2.23e3i)T^{2} \)
61 \( 1 + (-13.7 + 77.7i)T + (-3.49e3 - 1.27e3i)T^{2} \)
67 \( 1 + (84.5 - 70.9i)T + (779. - 4.42e3i)T^{2} \)
71 \( 1 + (-71.2 + 41.1i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (3.09 - 5.36i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-49.2 - 41.3i)T + (1.08e3 + 6.14e3i)T^{2} \)
83 \( 1 + (-97.6 + 116. i)T + (-1.19e3 - 6.78e3i)T^{2} \)
89 \( 1 + (-97.5 - 56.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (52.0 - 18.9i)T + (7.20e3 - 6.04e3i)T^{2} \)
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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

−13.26363364093008458845354403808, −12.13933244881135796289351917696, −10.90828678692164474723102064879, −10.47569759454770921903079391882, −8.963085765104479786785902231871, −7.64462450985235843230506651849, −6.21088227212213106709898898365, −5.19601465855950358137024351748, −3.73738118984307203280689899575, −0.813815529805682718239646765904, 2.06816420905515874500524229693, 4.44732023113174156399353744956, 5.78169410846402643323307518877, 6.75830145337255066409488796330, 8.073950690217211056858875574411, 9.648354151810621401453523998081, 10.57674735612975540033698530857, 11.76718700903659784317414843484, 12.36301687870438558283784595319, 13.54304733112077027764754825392

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