Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.988 - 0.154i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.974 − 1.74i)2-s + (−0.586 + 2.94i)3-s + (−2.10 + 3.40i)4-s + (−1.26 − 7.19i)5-s + (5.71 − 1.84i)6-s + (−2.67 + 3.18i)7-s + (7.99 + 0.353i)8-s + (−8.31 − 3.45i)9-s + (−11.3 + 9.23i)10-s + (−18.4 − 3.24i)11-s + (−8.78 − 8.17i)12-s + (−15.1 − 5.50i)13-s + (8.17 + 1.56i)14-s + (21.9 + 0.490i)15-s + (−7.17 − 14.3i)16-s + (7.49 + 12.9i)17-s + ⋯
L(s)  = 1  + (−0.487 − 0.873i)2-s + (−0.195 + 0.980i)3-s + (−0.525 + 0.850i)4-s + (−0.253 − 1.43i)5-s + (0.951 − 0.306i)6-s + (−0.382 + 0.455i)7-s + (0.999 + 0.0441i)8-s + (−0.923 − 0.383i)9-s + (−1.13 + 0.923i)10-s + (−1.67 − 0.295i)11-s + (−0.731 − 0.681i)12-s + (−1.16 − 0.423i)13-s + (0.584 + 0.111i)14-s + (1.46 + 0.0327i)15-s + (−0.448 − 0.893i)16-s + (0.441 + 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.154i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.988 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.988 - 0.154i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.988 - 0.154i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0130941 + 0.168862i\)
\(L(\frac12)\)  \(\approx\)  \(0.0130941 + 0.168862i\)
\(L(2)\)   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 \( 1 + (0.974 + 1.74i)T \)
3 \( 1 + (0.586 - 2.94i)T \)
good5 \( 1 + (1.26 + 7.19i)T + (-23.4 + 8.55i)T^{2} \)
7 \( 1 + (2.67 - 3.18i)T + (-8.50 - 48.2i)T^{2} \)
11 \( 1 + (18.4 + 3.24i)T + (113. + 41.3i)T^{2} \)
13 \( 1 + (15.1 + 5.50i)T + (129. + 108. i)T^{2} \)
17 \( 1 + (-7.49 - 12.9i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-0.338 - 0.195i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (10.6 + 12.7i)T + (-91.8 + 520. i)T^{2} \)
29 \( 1 + (-36.9 + 13.4i)T + (644. - 540. i)T^{2} \)
31 \( 1 + (-14.0 - 16.6i)T + (-166. + 946. i)T^{2} \)
37 \( 1 + (-5.57 - 9.65i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (56.8 + 20.6i)T + (1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (24.4 + 4.31i)T + (1.73e3 + 632. i)T^{2} \)
47 \( 1 + (-14.6 + 17.4i)T + (-383. - 2.17e3i)T^{2} \)
53 \( 1 - 51.2T + 2.80e3T^{2} \)
59 \( 1 + (82.3 - 14.5i)T + (3.27e3 - 1.19e3i)T^{2} \)
61 \( 1 + (-19.0 - 15.9i)T + (646. + 3.66e3i)T^{2} \)
67 \( 1 + (-17.9 + 49.4i)T + (-3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (75.5 - 43.6i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-17.8 + 30.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-3.59 - 9.87i)T + (-4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (34.4 + 94.7i)T + (-5.27e3 + 4.42e3i)T^{2} \)
89 \( 1 + (7.29 - 12.6i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (1.45 - 8.25i)T + (-8.84e3 - 3.21e3i)T^{2} \)
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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

−12.45810478606031292568640319679, −12.06017457763442154352616227870, −10.47559204417429126279418003115, −9.929784639211855146955757471479, −8.678809345538832550362965968935, −8.072413387382374398276282811970, −5.39855108067011939944868256347, −4.54483709604087286151239593138, −2.86558464182175270881092349676, −0.14029238966193242504425531768, 2.66025384803463798176138704026, 5.18220479975669359652148970083, 6.62364091154963391653428477053, 7.30221111601300798214437242619, 7.985537260458693007451747360909, 9.889465522899868270062967073070, 10.60237687525288191098934919986, 11.88661146515179538580712560994, 13.31864384014145112406321118647, 14.07054263279029420115779819691

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