Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.866 - 0.5i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + 7·5-s − 8.66i·7-s + 7.99·8-s + (−7 + 12.1i)10-s + 8.66i·11-s + 20·13-s + (15 + 8.66i)14-s + (−8 + 13.8i)16-s − 8·17-s + 10.3i·19-s + (−13.9 − 24.2i)20-s + (−15 − 8.66i)22-s − 3.46i·23-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 1.40·5-s − 1.23i·7-s + 0.999·8-s + (−0.700 + 1.21i)10-s + 0.787i·11-s + 1.53·13-s + (1.07 + 0.618i)14-s + (−0.5 + 0.866i)16-s − 0.470·17-s + 0.546i·19-s + (−0.699 − 1.21i)20-s + (−0.681 − 0.393i)22-s − 0.150i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.866 - 0.5i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (55, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.866 - 0.5i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.23554 + 0.331063i\)
\(L(\frac12)\)  \(\approx\)  \(1.23554 + 0.331063i\)
\(L(2)\)   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 \( 1 + (1 - 1.73i)T \)
3 \( 1 \)
good5 \( 1 - 7T + 25T^{2} \)
7 \( 1 + 8.66iT - 49T^{2} \)
11 \( 1 - 8.66iT - 121T^{2} \)
13 \( 1 - 20T + 169T^{2} \)
17 \( 1 + 8T + 289T^{2} \)
19 \( 1 - 10.3iT - 361T^{2} \)
23 \( 1 + 3.46iT - 529T^{2} \)
29 \( 1 - 10T + 841T^{2} \)
31 \( 1 + 53.6iT - 961T^{2} \)
37 \( 1 + 10T + 1.36e3T^{2} \)
41 \( 1 + 50T + 1.68e3T^{2} \)
43 \( 1 - 17.3iT - 1.84e3T^{2} \)
47 \( 1 - 86.6iT - 2.20e3T^{2} \)
53 \( 1 + 47T + 2.80e3T^{2} \)
59 \( 1 + 34.6iT - 3.48e3T^{2} \)
61 \( 1 + 64T + 3.72e3T^{2} \)
67 \( 1 - 86.6iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 55T + 5.32e3T^{2} \)
79 \( 1 - 6.92iT - 6.24e3T^{2} \)
83 \( 1 - 29.4iT - 6.88e3T^{2} \)
89 \( 1 - 10T + 7.92e3T^{2} \)
97 \( 1 + 25T + 9.40e3T^{2} \)
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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

−13.65171717859994979819767824748, −13.11240047239666905369613040605, −10.95749229245044362711917195548, −10.12899020510223067369966124106, −9.323127331205507395306825099192, −8.003623767483553983288030209150, −6.72125585241181109642264742912, −5.88395037204140374377020683437, −4.33574628821866167767952593205, −1.48497583204864638617788270488, 1.72898840209495218283428661696, 3.14448217799822426958685406920, 5.28648611455279355077013166041, 6.45257186795273421476901028968, 8.654477570599061711113866632041, 8.919945722914774266455946141385, 10.23232057975926124365667829440, 11.15001896174840744393383480555, 12.24432476250521832099858364681, 13.41405183791542871506647639331

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