Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.574 - 0.818i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.81 + 1.20i)2-s + (13.0 − 9.19i)4-s − 13.3·5-s − 25.4i·7-s + (−38.8 + 50.8i)8-s + (51.0 − 16.1i)10-s + 34.7i·11-s + 116.·13-s + (30.6 + 96.9i)14-s + (86.9 − 240. i)16-s − 109.·17-s + 264. i·19-s + (−175. + 123. i)20-s + (−41.8 − 132. i)22-s + 890. i·23-s + ⋯
L(s)  = 1  + (−0.953 + 0.301i)2-s + (0.818 − 0.574i)4-s − 0.535·5-s − 0.518i·7-s + (−0.607 + 0.794i)8-s + (0.510 − 0.161i)10-s + 0.287i·11-s + 0.691·13-s + (0.156 + 0.494i)14-s + (0.339 − 0.940i)16-s − 0.379·17-s + 0.733i·19-s + (−0.438 + 0.307i)20-s + (−0.0865 − 0.273i)22-s + 1.68i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.574 - 0.818i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (55, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.574 - 0.818i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.249381 + 0.479811i\)
\(L(\frac12)\)  \(\approx\)  \(0.249381 + 0.479811i\)
\(L(3)\)   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 \( 1 + (3.81 - 1.20i)T \)
3 \( 1 \)
good5 \( 1 + 13.3T + 625T^{2} \)
7 \( 1 + 25.4iT - 2.40e3T^{2} \)
11 \( 1 - 34.7iT - 1.46e4T^{2} \)
13 \( 1 - 116.T + 2.85e4T^{2} \)
17 \( 1 + 109.T + 8.35e4T^{2} \)
19 \( 1 - 264. iT - 1.30e5T^{2} \)
23 \( 1 - 890. iT - 2.79e5T^{2} \)
29 \( 1 + 1.11e3T + 7.07e5T^{2} \)
31 \( 1 - 1.66e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.98e3T + 1.87e6T^{2} \)
41 \( 1 - 1.22e3T + 2.82e6T^{2} \)
43 \( 1 - 3.14e3iT - 3.41e6T^{2} \)
47 \( 1 + 2.36e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.18e3T + 7.89e6T^{2} \)
59 \( 1 + 1.80e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.51e3T + 1.38e7T^{2} \)
67 \( 1 - 5.25e3iT - 2.01e7T^{2} \)
71 \( 1 + 1.12e3iT - 2.54e7T^{2} \)
73 \( 1 + 6.35e3T + 2.83e7T^{2} \)
79 \( 1 + 445. iT - 3.89e7T^{2} \)
83 \( 1 - 1.11e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.03e4T + 6.27e7T^{2} \)
97 \( 1 - 474.T + 8.85e7T^{2} \)
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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

−13.43504812078684563553408609793, −11.97544055446868067102243298034, −11.08549729512021827744869167547, −10.10085318142154167005583120085, −8.966253056355317204236498630758, −7.84466958158795072318236356346, −6.98420943574439811030621433309, −5.56756710223584055562149441122, −3.66050408114507539204266642193, −1.51219220201182143335381569914, 0.33288481037541383349117351049, 2.31732638318030288464284135470, 3.91525283333979106170886721587, 5.98724996292795817032274003905, 7.28973781836994899248290213114, 8.451699046053884693291304935888, 9.199751803935128032668647631474, 10.59697906836921970559591672693, 11.40183335764694420567424535394, 12.31968970003056443931629667833

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