Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.0394 + 0.999i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.63 − 1.15i)2-s + (1.32 − 3.77i)4-s + (−3.07 − 5.32i)5-s + (−0.511 − 0.295i)7-s + (−2.20 − 7.68i)8-s + (−11.1 − 5.12i)10-s + (15.1 + 8.72i)11-s + (−0.892 − 1.54i)13-s + (−1.17 + 0.110i)14-s + (−12.5 − 9.98i)16-s + 16.9·17-s + 19.5i·19-s + (−24.1 + 4.56i)20-s + (34.7 − 3.25i)22-s + (−6.86 + 3.96i)23-s + ⋯
L(s)  = 1  + (0.815 − 0.578i)2-s + (0.330 − 0.943i)4-s + (−0.614 − 1.06i)5-s + (−0.0730 − 0.0421i)7-s + (−0.276 − 0.961i)8-s + (−1.11 − 0.512i)10-s + (1.37 + 0.793i)11-s + (−0.0686 − 0.118i)13-s + (−0.0840 + 0.00785i)14-s + (−0.781 − 0.624i)16-s + 0.995·17-s + 1.02i·19-s + (−1.20 + 0.228i)20-s + (1.58 − 0.147i)22-s + (−0.298 + 0.172i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0394 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0394 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.0394 + 0.999i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.0394 + 0.999i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.33669 - 1.39048i\)
\(L(\frac12)\)  \(\approx\)  \(1.33669 - 1.39048i\)
\(L(2)\)   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 \( 1 + (-1.63 + 1.15i)T \)
3 \( 1 \)
good5 \( 1 + (3.07 + 5.32i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (0.511 + 0.295i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-15.1 - 8.72i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (0.892 + 1.54i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 16.9T + 289T^{2} \)
19 \( 1 - 19.5iT - 361T^{2} \)
23 \( 1 + (6.86 - 3.96i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (3.17 - 5.49i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (27.6 - 15.9i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 58.2T + 1.36e3T^{2} \)
41 \( 1 + (-2.66 - 4.62i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-33.9 - 19.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (9.64 + 5.56i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 35.8T + 2.80e3T^{2} \)
59 \( 1 + (20.8 - 12.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (37.9 - 65.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-31.8 + 18.3i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 87.8iT - 5.04e3T^{2} \)
73 \( 1 + 60.0T + 5.32e3T^{2} \)
79 \( 1 + (32.1 + 18.5i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (66.0 + 38.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 27.5T + 7.92e3T^{2} \)
97 \( 1 + (-13.0 + 22.6i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.84619882282114889423859046055, −12.26537412675680086680474033361, −11.53273532674068489132282327943, −10.05666080128007655945758818687, −9.108387197351374955973595071113, −7.58229044112217861292914952631, −6.04671359275668926567694579432, −4.68218835327414948479918632447, −3.68954253217821728205388211764, −1.37514981626963821634744133611, 3.04559944863135219147735352834, 4.12554892505450450896174339640, 5.92064358389547114739783893614, 6.88678385622481135964430242834, 7.88093530018248624103139717257, 9.273281323722127479249224805579, 11.08309201174809340862783054849, 11.60097160418020752293400833062, 12.78028534563087792915008123396, 14.11706453805308714802494871324

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