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Properties

Label 2-3e5-3.2-c8-0-27
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  + 12.7i·2-s + 92.2·4-s − 113. i·5-s + 2.73e3·7-s + 4.45e3i·8-s + 1.44e3·10-s + 1.68e4i·11-s − 5.58e3·13-s + 3.50e4i·14-s − 3.33e4·16-s + 9.97e4i·17-s − 2.34e5·19-s − 1.04e4i·20-s − 2.15e5·22-s − 4.23e5i·23-s + ⋯
L(s)  = 1  + 0.799i·2-s + 0.360·4-s − 0.180i·5-s + 1.14·7-s + 1.08i·8-s + 0.144·10-s + 1.15i·11-s − 0.195·13-s + 0.911i·14-s − 0.509·16-s + 1.19i·17-s − 1.79·19-s − 0.0652i·20-s − 0.921·22-s − 1.51i·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

\(L(\frac{9}{2})\) \(\approx\) \(2.009379818\)
\(L(\frac12)\) \(\approx\) \(2.009379818\)
\(L(5)\) 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 \( 1 - 12.7iT - 256T^{2} \)
5 \( 1 + 113. iT - 3.90e5T^{2} \)
7 \( 1 - 2.73e3T + 5.76e6T^{2} \)
11 \( 1 - 1.68e4iT - 2.14e8T^{2} \)
13 \( 1 + 5.58e3T + 8.15e8T^{2} \)
17 \( 1 - 9.97e4iT - 6.97e9T^{2} \)
19 \( 1 + 2.34e5T + 1.69e10T^{2} \)
23 \( 1 + 4.23e5iT - 7.83e10T^{2} \)
29 \( 1 - 6.79e5iT - 5.00e11T^{2} \)
31 \( 1 + 4.51e5T + 8.52e11T^{2} \)
37 \( 1 + 5.23e5T + 3.51e12T^{2} \)
41 \( 1 - 5.06e5iT - 7.98e12T^{2} \)
43 \( 1 + 6.59e5T + 1.16e13T^{2} \)
47 \( 1 + 4.80e6iT - 2.38e13T^{2} \)
53 \( 1 - 1.33e7iT - 6.22e13T^{2} \)
59 \( 1 + 4.89e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.05e7T + 1.91e14T^{2} \)
67 \( 1 - 2.36e7T + 4.06e14T^{2} \)
71 \( 1 - 3.21e7iT - 6.45e14T^{2} \)
73 \( 1 + 2.57e7T + 8.06e14T^{2} \)
79 \( 1 - 2.79e7T + 1.51e15T^{2} \)
83 \( 1 + 4.94e7iT - 2.25e15T^{2} \)
89 \( 1 - 1.14e8iT - 3.93e15T^{2} \)
97 \( 1 + 8.78e7T + 7.83e15T^{2} \)
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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

−10.91710400052699381102980697274, −10.47612141642514597741482791615, −8.740588063380440022991694552896, −8.218722114459588490295837697489, −7.13256305815912085133063218699, −6.34510395744058689384762186851, −5.07760646436407753917265504340, −4.32035158005918287581219433530, −2.35429214472428746871835876485, −1.57115339231693553132439984038, 0.38400386390156033206478195775, 1.54562389758466102689093871842, 2.53913972032252113666431008386, 3.63485509968748151242191040303, 4.89228421262671958059477943368, 6.15395018327773890021941026320, 7.26800515840435133794452193012, 8.284663537335515695792017211384, 9.369952312006999592600681976936, 10.55393948762741689179724044969

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