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  :  $\chi_{108} (71, \cdot )$
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

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

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