Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} $
Sign $0.883 - 0.467i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.32 + 0.5i)2-s + (1.50 + 1.32i)4-s i·5-s + 7-s + (1.32 + 2.50i)8-s + (0.5 − 1.32i)10-s − 3i·11-s + 5.29i·13-s + (1.32 + 0.5i)14-s + (0.500 + 3.96i)16-s − 5.29·17-s − 5.29i·19-s + (1.32 − 1.50i)20-s + (1.5 − 3.96i)22-s − 5.29·23-s + ⋯
L(s)  = 1  + (0.935 + 0.353i)2-s + (0.750 + 0.661i)4-s − 0.447i·5-s + 0.377·7-s + (0.467 + 0.883i)8-s + (0.158 − 0.418i)10-s − 0.904i·11-s + 1.46i·13-s + (0.353 + 0.133i)14-s + (0.125 + 0.992i)16-s − 1.28·17-s − 1.21i·19-s + (0.295 − 0.335i)20-s + (0.319 − 0.846i)22-s − 1.10·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(216\)    =    \(2^{3} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.883 - 0.467i$
motivic weight  =  \(1\)
character  :  $\chi_{216} (109, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 216,\ (\ :1/2),\ 0.883 - 0.467i)$
$L(1)$  $\approx$  $1.98160 + 0.491968i$
$L(\frac12)$  $\approx$  $1.98160 + 0.491968i$
$L(\frac{3}{2})$   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 \( 1 + (-1.32 - 0.5i)T \)
3 \( 1 \)
good5 \( 1 + iT - 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 - 5.29iT - 13T^{2} \)
17 \( 1 + 5.29T + 17T^{2} \)
19 \( 1 + 5.29iT - 19T^{2} \)
23 \( 1 + 5.29T + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 - 5.29iT - 37T^{2} \)
41 \( 1 - 5.29T + 41T^{2} \)
43 \( 1 + 10.5iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 15.8T + 71T^{2} \)
73 \( 1 - 3T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 7iT - 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−12.49156053404005481847778531537, −11.48080931230343463067942220562, −10.97670246038403154801934236230, −9.172258595227955719917877285612, −8.384929088395324122507493186650, −7.08058733960919291030355702453, −6.16423515543451171126966433786, −4.88329052261943764743593500737, −4.00212274472776714422021322012, −2.24793440843523719707062138465, 2.02578688338605839856043544083, 3.46551182701967147505360841496, 4.74187021437915437324284521934, 5.84368210366236991635008430520, 6.98990989642052134622472713068, 8.034773514306085629413047787340, 9.667227743779681612511191925638, 10.61021412429199440223111872084, 11.23031609078296604504597801944, 12.60030423381518539387494534213

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