Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.596 - 0.802i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.889 − 2.68i)2-s + (−6.41 − 4.77i)4-s + 14.9i·5-s + 30.0i·7-s + (−18.5 + 12.9i)8-s + (40.0 + 13.2i)10-s − 55.9·11-s + 57.4·13-s + (80.6 + 26.7i)14-s + (18.3 + 61.2i)16-s − 29.2i·17-s − 0.709i·19-s + (71.2 − 95.7i)20-s + (−49.7 + 150. i)22-s − 48.0·23-s + ⋯
L(s)  = 1  + (0.314 − 0.949i)2-s + (−0.802 − 0.596i)4-s + 1.33i·5-s + 1.62i·7-s + (−0.818 + 0.573i)8-s + (1.26 + 0.419i)10-s − 1.53·11-s + 1.22·13-s + (1.54 + 0.510i)14-s + (0.287 + 0.957i)16-s − 0.417i·17-s − 0.00856i·19-s + (0.796 − 1.07i)20-s + (−0.482 + 1.45i)22-s − 0.435·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.596 - 0.802i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 0.596 - 0.802i)\)
\(L(2)\)  \(\approx\)  \(1.13683 + 0.571166i\)
\(L(\frac12)\)  \(\approx\)  \(1.13683 + 0.571166i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-0.889 + 2.68i)T \)
3 \( 1 \)
good5 \( 1 - 14.9iT - 125T^{2} \)
7 \( 1 - 30.0iT - 343T^{2} \)
11 \( 1 + 55.9T + 1.33e3T^{2} \)
13 \( 1 - 57.4T + 2.19e3T^{2} \)
17 \( 1 + 29.2iT - 4.91e3T^{2} \)
19 \( 1 + 0.709iT - 6.85e3T^{2} \)
23 \( 1 + 48.0T + 1.21e4T^{2} \)
29 \( 1 - 172. iT - 2.43e4T^{2} \)
31 \( 1 - 45.2iT - 2.97e4T^{2} \)
37 \( 1 - 248.T + 5.06e4T^{2} \)
41 \( 1 + 51.3iT - 6.89e4T^{2} \)
43 \( 1 - 19.9iT - 7.95e4T^{2} \)
47 \( 1 - 10.8T + 1.03e5T^{2} \)
53 \( 1 - 37.0iT - 1.48e5T^{2} \)
59 \( 1 + 411.T + 2.05e5T^{2} \)
61 \( 1 + 308.T + 2.26e5T^{2} \)
67 \( 1 - 113. iT - 3.00e5T^{2} \)
71 \( 1 - 1.13e3T + 3.57e5T^{2} \)
73 \( 1 - 728.T + 3.89e5T^{2} \)
79 \( 1 - 487. iT - 4.93e5T^{2} \)
83 \( 1 - 1.16e3T + 5.71e5T^{2} \)
89 \( 1 + 1.19e3iT - 7.04e5T^{2} \)
97 \( 1 - 624.T + 9.12e5T^{2} \)
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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

−13.24525307163752104248352608960, −12.25585945582470939571015475747, −11.16156934748492069304165660059, −10.56210660933292032035838577650, −9.297339001163552815935185371277, −8.123842192207860293186957006688, −6.25626710135042528484280788531, −5.25358146380838650082198008324, −3.22388443119844686436524485819, −2.34727550402845558925190920252, 0.64194721744895297822515307471, 3.88495086884369451655871156996, 4.86759391993405853381390285315, 6.15004715303952178088390907009, 7.72491799805886435140260142927, 8.265604842201450830814830070186, 9.661206126872721671954988035691, 10.88023070015445938752189535116, 12.55215893325773420364787256426, 13.38755273534483625848827607945

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