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Properties

Label 2-3e5-3.2-c8-0-50
Degree $2$
Conductor $243$
Sign $-1$
Analytic cond. $98.9930$
Root an. cond. $9.94952$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 30.7i·2-s − 691.·4-s + 787. i·5-s + 3.93e3·7-s − 1.34e4i·8-s − 2.42e4·10-s − 6.82e3i·11-s − 2.03e4·13-s + 1.21e5i·14-s + 2.35e5·16-s − 8.13e4i·17-s + 1.52e5·19-s − 5.44e5i·20-s + 2.10e5·22-s − 2.18e5i·23-s + ⋯
L(s)  = 1  + 1.92i·2-s − 2.70·4-s + 1.25i·5-s + 1.63·7-s − 3.27i·8-s − 2.42·10-s − 0.466i·11-s − 0.711·13-s + 3.15i·14-s + 3.59·16-s − 0.974i·17-s + 1.16·19-s − 3.40i·20-s + 0.897·22-s − 0.781i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $-1$
Analytic conductor: \(98.9930\)
Root analytic conductor: \(9.94952\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 243,\ (\ :4),\ -1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.261646785\)
\(L(\frac12)\) \(\approx\) \(2.261646785\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 - 30.7iT - 256T^{2} \)
5 \( 1 - 787. iT - 3.90e5T^{2} \)
7 \( 1 - 3.93e3T + 5.76e6T^{2} \)
11 \( 1 + 6.82e3iT - 2.14e8T^{2} \)
13 \( 1 + 2.03e4T + 8.15e8T^{2} \)
17 \( 1 + 8.13e4iT - 6.97e9T^{2} \)
19 \( 1 - 1.52e5T + 1.69e10T^{2} \)
23 \( 1 + 2.18e5iT - 7.83e10T^{2} \)
29 \( 1 - 6.30e5iT - 5.00e11T^{2} \)
31 \( 1 - 1.73e6T + 8.52e11T^{2} \)
37 \( 1 - 2.10e5T + 3.51e12T^{2} \)
41 \( 1 + 8.46e5iT - 7.98e12T^{2} \)
43 \( 1 + 6.45e5T + 1.16e13T^{2} \)
47 \( 1 - 8.25e6iT - 2.38e13T^{2} \)
53 \( 1 - 3.70e6iT - 6.22e13T^{2} \)
59 \( 1 - 1.54e7iT - 1.46e14T^{2} \)
61 \( 1 - 4.12e6T + 1.91e14T^{2} \)
67 \( 1 - 1.54e6T + 4.06e14T^{2} \)
71 \( 1 + 3.41e7iT - 6.45e14T^{2} \)
73 \( 1 - 4.41e7T + 8.06e14T^{2} \)
79 \( 1 + 2.48e6T + 1.51e15T^{2} \)
83 \( 1 + 4.22e7iT - 2.25e15T^{2} \)
89 \( 1 - 4.68e7iT - 3.93e15T^{2} \)
97 \( 1 - 4.86e7T + 7.83e15T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.01979211649106347221308125134, −9.926883373566693648686522849908, −8.794669847560681655295971536846, −7.80864978212664691631814225950, −7.30318829378763344092277836158, −6.35463439010129186231298463506, −5.21164100280408180858402419505, −4.52598676293045316454817427635, −2.92037204310748838643333019874, −0.860709589528603839835316965160, 0.73115958801928298304423624265, 1.45335534407948283845502377110, 2.28440606531217541076731091968, 3.88234444113903362170118364619, 4.82146942972927487756357728483, 5.24132288722289403350009216390, 7.938729649721365826738589508070, 8.455246067397996685430323829177, 9.536179665925641501251612716082, 10.22934017628560951396755170717

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