Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.250 - 0.968i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.823 + 2.70i)2-s + (−6.64 − 4.45i)4-s + (4.71 − 2.72i)5-s + (20.9 + 12.0i)7-s + (17.5 − 14.3i)8-s + (3.48 + 14.9i)10-s + (−25.3 + 43.9i)11-s + (25.0 + 43.4i)13-s + (−49.9 + 46.6i)14-s + (24.2 + 59.2i)16-s − 51.7i·17-s + 27.9i·19-s + (−43.4 − 2.93i)20-s + (−98.1 − 104. i)22-s + (3.93 + 6.81i)23-s + ⋯
L(s)  = 1  + (−0.291 + 0.956i)2-s + (−0.830 − 0.557i)4-s + (0.421 − 0.243i)5-s + (1.13 + 0.652i)7-s + (0.774 − 0.632i)8-s + (0.110 + 0.474i)10-s + (−0.696 + 1.20i)11-s + (0.535 + 0.927i)13-s + (−0.953 + 0.891i)14-s + (0.379 + 0.925i)16-s − 0.737i·17-s + 0.337i·19-s + (−0.485 − 0.0328i)20-s + (−0.950 − 1.01i)22-s + (0.0356 + 0.0617i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.250 - 0.968i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.250 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.250 - 0.968i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (35, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.250 - 0.968i)\)
\(L(2)\)  \(\approx\)  \(0.865087 + 1.11784i\)
\(L(\frac12)\)  \(\approx\)  \(0.865087 + 1.11784i\)
\(L(\frac{5}{2})\)   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 \( 1 + (0.823 - 2.70i)T \)
3 \( 1 \)
good5 \( 1 + (-4.71 + 2.72i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (-20.9 - 12.0i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (25.3 - 43.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-25.0 - 43.4i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 51.7iT - 4.91e3T^{2} \)
19 \( 1 - 27.9iT - 6.85e3T^{2} \)
23 \( 1 + (-3.93 - 6.81i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-212. - 122. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (51.4 - 29.6i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 295.T + 5.06e4T^{2} \)
41 \( 1 + (-146. + 84.7i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (284. + 164. i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-47.9 + 83.0i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 300. iT - 1.48e5T^{2} \)
59 \( 1 + (113. + 196. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-173. + 300. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-904. + 522. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 243.T + 3.57e5T^{2} \)
73 \( 1 + 1.09e3T + 3.89e5T^{2} \)
79 \( 1 + (530. + 306. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (283. - 490. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 212. iT - 7.04e5T^{2} \)
97 \( 1 + (-234. + 405. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.82382156163867269270851090230, −12.66986509227701409136258065486, −11.38942707612594099789562415271, −10.01217766922963030428574293660, −9.037732753382994869957665709850, −8.045133523556399068069319544192, −6.90524950823750360032620783257, −5.44732773363148190940315204094, −4.61252328122639752805582735762, −1.76078645700658011303519445294, 0.973865684584080583019039752334, 2.77445691046258832668078341724, 4.34970949112923483700975551791, 5.82602566705033067656279925626, 7.892857509013054702820587457867, 8.461654592260947560925550125557, 10.10477630632544917737743351845, 10.79189543854353646245034503358, 11.53897563188113892126592278214, 13.02381865679441366135049945346

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