Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.967 + 0.252i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.87 + 2.87i)2-s + (15.5 − 27.9i)4-s + (−70.1 + 40.5i)5-s + (−89.9 − 51.9i)7-s + (4.79 + 180. i)8-s + (225. − 399. i)10-s + (−214. + 371. i)11-s + (42.9 + 74.4i)13-s + (587. − 5.18i)14-s + (−542. − 868. i)16-s − 1.50e3i·17-s + 1.22e3i·19-s + (45.7 + 2.59e3i)20-s + (−21.3 − 2.42e3i)22-s + (−1.53e3 − 2.65e3i)23-s + ⋯
L(s)  = 1  + (−0.861 + 0.507i)2-s + (0.484 − 0.874i)4-s + (−1.25 + 0.724i)5-s + (−0.694 − 0.400i)7-s + (0.0264 + 0.999i)8-s + (0.713 − 1.26i)10-s + (−0.533 + 0.924i)11-s + (0.0704 + 0.122i)13-s + (0.801 − 0.00707i)14-s + (−0.530 − 0.847i)16-s − 1.26i·17-s + 0.778i·19-s + (0.0255 + 1.44i)20-s + (−0.00942 − 1.06i)22-s + (−0.603 − 1.04i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.967 + 0.252i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (35, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ 0.967 + 0.252i)\)
\(L(3)\)  \(\approx\)  \(0.538800 - 0.0690222i\)
\(L(\frac12)\)  \(\approx\)  \(0.538800 - 0.0690222i\)
\(L(\frac{7}{2})\)   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 \( 1 + (4.87 - 2.87i)T \)
3 \( 1 \)
good5 \( 1 + (70.1 - 40.5i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (89.9 + 51.9i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (214. - 371. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-42.9 - 74.4i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + 1.50e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.22e3iT - 2.47e6T^{2} \)
23 \( 1 + (1.53e3 + 2.65e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-4.34e3 - 2.50e3i)T + (1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-3.40e3 + 1.96e3i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 1.30e4T + 6.93e7T^{2} \)
41 \( 1 + (-1.50e4 + 8.70e3i)T + (5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (-5.75e3 - 3.32e3i)T + (7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (1.00e4 - 1.74e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + 1.80e4iT - 4.18e8T^{2} \)
59 \( 1 + (1.47e4 + 2.55e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.19e4 - 2.07e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-348. + 201. i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 3.66e4T + 1.80e9T^{2} \)
73 \( 1 - 5.86e4T + 2.07e9T^{2} \)
79 \( 1 + (3.93e4 + 2.27e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-2.72e4 + 4.72e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 4.53e4iT - 5.58e9T^{2} \)
97 \( 1 + (-4.18e4 + 7.24e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−12.50038264421952020494325925674, −11.43108847796534056346366894808, −10.42974218150812148801379748221, −9.568528275681722149160693142651, −8.032139309555481188246525640047, −7.33284208780981539054535571600, −6.37502963994269983217407874542, −4.48395520897605066703552601032, −2.73997009334611429479280518417, −0.41122560949418696984098717153, 0.809229191078705717135241269034, 2.93212043431979688312495294594, 4.16682826261890807709537787983, 6.15549951087410329074209966523, 7.79580416593490963191671700846, 8.404617009416195075010362588392, 9.469719231041476276960655460369, 10.77134094934101918192738816257, 11.68842559626300589357889448460, 12.53785887924850297387937928081

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