Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.984 - 0.175i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.86 + 1.03i)2-s + (13.8 + 7.96i)4-s + (5.89 − 10.2i)5-s + (50.5 − 29.1i)7-s + (45.4 + 45.0i)8-s + (33.2 − 33.3i)10-s + (−86.9 + 50.2i)11-s + (85.3 − 147. i)13-s + (225. − 60.7i)14-s + (129. + 221. i)16-s + 398.·17-s + 404. i·19-s + (163. − 94.7i)20-s + (−387. + 104. i)22-s + (−291. − 168. i)23-s + ⋯
L(s)  = 1  + (0.966 + 0.257i)2-s + (0.867 + 0.497i)4-s + (0.235 − 0.408i)5-s + (1.03 − 0.595i)7-s + (0.709 + 0.704i)8-s + (0.332 − 0.333i)10-s + (−0.718 + 0.414i)11-s + (0.504 − 0.874i)13-s + (1.15 − 0.309i)14-s + (0.504 + 0.863i)16-s + 1.37·17-s + 1.12i·19-s + (0.407 − 0.236i)20-s + (−0.801 + 0.215i)22-s + (−0.550 − 0.317i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.984 - 0.175i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (91, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.984 - 0.175i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(3.56286 + 0.314559i\)
\(L(\frac12)\)  \(\approx\)  \(3.56286 + 0.314559i\)
\(L(3)\)   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 \( 1 + (-3.86 - 1.03i)T \)
3 \( 1 \)
good5 \( 1 + (-5.89 + 10.2i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (-50.5 + 29.1i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (86.9 - 50.2i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-85.3 + 147. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 398.T + 8.35e4T^{2} \)
19 \( 1 - 404. iT - 1.30e5T^{2} \)
23 \( 1 + (291. + 168. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (327. + 567. i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (550. + 317. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 1.59e3T + 1.87e6T^{2} \)
41 \( 1 + (-1.23e3 + 2.13e3i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (1.93e3 - 1.11e3i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (2.51e3 - 1.45e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 1.29e3T + 7.89e6T^{2} \)
59 \( 1 + (-1.00e3 - 578. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (2.96e3 + 5.12e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-3.08e3 - 1.78e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 5.63e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.49e3T + 2.83e7T^{2} \)
79 \( 1 + (2.78e3 - 1.61e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (7.06e3 - 4.07e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 910.T + 6.27e7T^{2} \)
97 \( 1 + (-8.80e3 - 1.52e4i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−13.04163477155986209579646648462, −12.26773139561610858823490976902, −11.04217359750718145921797652207, −10.12458163343471900081587472489, −8.137346564256624848790978941657, −7.57071473609169803671639172276, −5.81223482390554199655394053193, −4.95201654831335493265335767807, −3.54265675135989815109260643119, −1.61865883687529997469984674738, 1.72363296000423943215388115230, 3.15903311004024287898937419989, 4.83424756743203083161320826458, 5.80667268502240043797833223234, 7.14882810172898967747274088862, 8.519190213159112859871642888892, 10.10651639671999949284062490699, 11.16284308015116388840899935454, 11.83359642519509230255402221091, 13.03690389588756930847973141556

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