Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.393 + 0.919i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.49 − 2.59i)3-s + (−5.00 − 5.96i)5-s + (−3.39 + 1.23i)7-s + (−4.51 − 7.78i)9-s + (2.59 − 3.09i)11-s + (2.31 + 13.1i)13-s + (−22.9 + 4.07i)15-s + (20.7 − 11.9i)17-s + (13.5 − 23.5i)19-s + (−1.87 + 10.6i)21-s + (3.97 − 10.9i)23-s + (−6.17 + 35.0i)25-s + (−26.9 + 0.0701i)27-s + (22.8 + 4.03i)29-s + (−3.81 − 1.38i)31-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)3-s + (−1.00 − 1.19i)5-s + (−0.484 + 0.176i)7-s + (−0.501 − 0.865i)9-s + (0.236 − 0.281i)11-s + (0.177 + 1.00i)13-s + (−1.53 + 0.271i)15-s + (1.22 − 0.705i)17-s + (0.715 − 1.23i)19-s + (−0.0891 + 0.508i)21-s + (0.172 − 0.474i)23-s + (−0.247 + 1.40i)25-s + (−0.999 + 0.00259i)27-s + (0.788 + 0.139i)29-s + (−0.122 − 0.0447i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.393 + 0.919i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (29, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.393 + 0.919i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.668171 - 1.01303i\)
\(L(\frac12)\)  \(\approx\)  \(0.668171 - 1.01303i\)
\(L(2)\)   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 \( 1 \)
3 \( 1 + (-1.49 + 2.59i)T \)
good5 \( 1 + (5.00 + 5.96i)T + (-4.34 + 24.6i)T^{2} \)
7 \( 1 + (3.39 - 1.23i)T + (37.5 - 31.4i)T^{2} \)
11 \( 1 + (-2.59 + 3.09i)T + (-21.0 - 119. i)T^{2} \)
13 \( 1 + (-2.31 - 13.1i)T + (-158. + 57.8i)T^{2} \)
17 \( 1 + (-20.7 + 11.9i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-13.5 + 23.5i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-3.97 + 10.9i)T + (-405. - 340. i)T^{2} \)
29 \( 1 + (-22.8 - 4.03i)T + (790. + 287. i)T^{2} \)
31 \( 1 + (3.81 + 1.38i)T + (736. + 617. i)T^{2} \)
37 \( 1 + (-35.3 - 61.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (43.0 - 7.59i)T + (1.57e3 - 574. i)T^{2} \)
43 \( 1 + (-35.3 - 29.6i)T + (321. + 1.82e3i)T^{2} \)
47 \( 1 + (28.3 + 77.9i)T + (-1.69e3 + 1.41e3i)T^{2} \)
53 \( 1 - 28.9iT - 2.80e3T^{2} \)
59 \( 1 + (-33.8 - 40.3i)T + (-604. + 3.42e3i)T^{2} \)
61 \( 1 + (-4.08 + 1.48i)T + (2.85e3 - 2.39e3i)T^{2} \)
67 \( 1 + (22.7 + 129. i)T + (-4.21e3 + 1.53e3i)T^{2} \)
71 \( 1 + (60.4 - 34.8i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-65.4 + 113. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (4.20 - 23.8i)T + (-5.86e3 - 2.13e3i)T^{2} \)
83 \( 1 + (41.0 + 7.22i)T + (6.47e3 + 2.35e3i)T^{2} \)
89 \( 1 + (-84.1 - 48.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-37.2 - 31.2i)T + (1.63e3 + 9.26e3i)T^{2} \)
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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

−13.10939165060512424303712914992, −12.02669291921544577086721201129, −11.64518264710203563932980374834, −9.486364898975401143119806270318, −8.701830375792773094373620391795, −7.71556549224761937636669217687, −6.54931846312790353671870329211, −4.81138177985495826538923054141, −3.20970635679828232100566091635, −0.916257868536647717022641158336, 3.14675159683883574619936611504, 3.84756548347869232459950156427, 5.75161616647380786549936193653, 7.41042698453308917175166183594, 8.180380917734395335743490203612, 9.837300535591430965601341652866, 10.44936093756566366195793870455, 11.49447715032646156125253989166, 12.70758362323519789852082209903, 14.26062573426865319625253575478

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