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Properties

Label 2-3e5-3.2-c8-0-34
Degree $2$
Conductor $243$
Sign $-i$
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  + 4.69i·2-s + 233.·4-s + 365. i·5-s − 3.78e3·7-s + 2.30e3i·8-s − 1.71e3·10-s + 1.30e4i·11-s + 2.13e4·13-s − 1.77e4i·14-s + 4.90e4·16-s − 1.48e5i·17-s + 1.92e5·19-s + 8.55e4i·20-s − 6.12e4·22-s − 5.16e5i·23-s + ⋯
L(s)  = 1  + 0.293i·2-s + 0.913·4-s + 0.585i·5-s − 1.57·7-s + 0.561i·8-s − 0.171·10-s + 0.891i·11-s + 0.748·13-s − 0.462i·14-s + 0.749·16-s − 1.77i·17-s + 1.47·19-s + 0.534i·20-s − 0.261·22-s − 1.84i·23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $-i$
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: \((2,\ 243,\ (\ :4),\ -i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.378230806\)
\(L(\frac12)\) \(\approx\) \(2.378230806\)
\(L(5)\) 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 \( 1 - 4.69iT - 256T^{2} \)
5 \( 1 - 365. iT - 3.90e5T^{2} \)
7 \( 1 + 3.78e3T + 5.76e6T^{2} \)
11 \( 1 - 1.30e4iT - 2.14e8T^{2} \)
13 \( 1 - 2.13e4T + 8.15e8T^{2} \)
17 \( 1 + 1.48e5iT - 6.97e9T^{2} \)
19 \( 1 - 1.92e5T + 1.69e10T^{2} \)
23 \( 1 + 5.16e5iT - 7.83e10T^{2} \)
29 \( 1 - 8.98e5iT - 5.00e11T^{2} \)
31 \( 1 - 5.57e5T + 8.52e11T^{2} \)
37 \( 1 - 3.28e5T + 3.51e12T^{2} \)
41 \( 1 - 4.45e6iT - 7.98e12T^{2} \)
43 \( 1 + 2.33e6T + 1.16e13T^{2} \)
47 \( 1 - 4.92e6iT - 2.38e13T^{2} \)
53 \( 1 - 1.52e6iT - 6.22e13T^{2} \)
59 \( 1 - 5.92e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.85e7T + 1.91e14T^{2} \)
67 \( 1 + 2.92e7T + 4.06e14T^{2} \)
71 \( 1 + 6.92e6iT - 6.45e14T^{2} \)
73 \( 1 - 3.58e7T + 8.06e14T^{2} \)
79 \( 1 - 1.99e5T + 1.51e15T^{2} \)
83 \( 1 - 1.96e7iT - 2.25e15T^{2} \)
89 \( 1 - 1.04e8iT - 3.93e15T^{2} \)
97 \( 1 - 3.06e7T + 7.83e15T^{2} \)
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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

−10.83377568712352053544982616331, −9.995350065176521122928051129807, −9.129609541418014389813198420735, −7.60254054746709161756209592089, −6.78538224387040803261777955727, −6.33407909651441216947355891255, −4.91542154798202377052332448148, −3.10813249476630465303828933157, −2.75608143693618737124328137789, −0.997635649236857076185991349762, 0.58559407946864221578226356593, 1.61747017623887916870353235432, 3.19950523995442721445135639600, 3.66551704707614882849519710132, 5.72338190067909685793571482281, 6.22426900992982886113221963665, 7.40119904460606863194614447432, 8.568902091108516646924935655771, 9.624712306014727031156274003653, 10.41395638835249691076954539520

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