Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.993 + 0.110i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.186 − 1.99i)2-s + (−3.93 − 0.741i)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 + (0.683 − 0.963i)14-s + (14.8 + 5.83i)16-s + 16.9·17-s − 19.5i·19-s + (8.13 + 23.2i)20-s + (−20.2 + 28.4i)22-s + (6.86 − 3.96i)23-s + ⋯
L(s)  = 1  + (0.0931 − 0.995i)2-s + (−0.982 − 0.185i)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 + (0.0488 − 0.0688i)14-s + (0.931 + 0.364i)16-s + 0.995·17-s − 1.02i·19-s + (0.406 + 1.16i)20-s + (−0.918 + 1.29i)22-s + (0.298 − 0.172i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.110i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.993 + 0.110i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.993 + 0.110i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0490842 - 0.882838i\)
\(L(\frac12)\)  \(\approx\)  \(0.0490842 - 0.882838i\)
\(L(2)\)   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 \( 1 + (-0.186 + 1.99i)T \)
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

−12.91888745725116598165573031473, −11.92214015985464820714954992159, −11.01715053347548368724976347255, −9.876840900504361444898969961019, −8.636588961698806219412128260553, −7.899293400097362324718808981778, −5.52093997716917428650586005268, −4.54321107208873278982535913771, −2.92745591367415112590191307043, −0.65264821412231623908511162464, 3.22520660201977415540696030214, 4.79356504430288030452703278670, 6.19650410086781981319547196527, 7.51018370075326456149083773266, 7.951312949550349412807386399363, 9.719063912891518120792281083344, 10.60554900792212034184051530432, 12.06591628335374293068147561021, 13.10168202050724836755918643534, 14.31149214762831894086044567053

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