Properties

Label 2-3e5-1.1-c1-0-5
Degree $2$
Conductor $243$
Sign $-1$
Analytic cond. $1.94036$
Root an. cond. $1.39296$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.53·2-s + 4.41·4-s − 0.467·5-s − 3.22·7-s − 6.10·8-s + 1.18·10-s + 3.10·11-s − 2.18·13-s + 8.17·14-s + 6.63·16-s − 3·17-s + 0.0418·19-s − 2.06·20-s − 7.86·22-s − 6.10·23-s − 4.78·25-s + 5.53·26-s − 14.2·28-s − 6.57·29-s − 6.22·31-s − 4.59·32-s + 7.59·34-s + 1.50·35-s + 3.59·37-s − 0.106·38-s + 2.85·40-s − 7.70·41-s + ⋯
L(s)  = 1  − 1.79·2-s + 2.20·4-s − 0.209·5-s − 1.21·7-s − 2.15·8-s + 0.374·10-s + 0.936·11-s − 0.605·13-s + 2.18·14-s + 1.65·16-s − 0.727·17-s + 0.00961·19-s − 0.461·20-s − 1.67·22-s − 1.27·23-s − 0.956·25-s + 1.08·26-s − 2.69·28-s − 1.22·29-s − 1.11·31-s − 0.812·32-s + 1.30·34-s + 0.255·35-s + 0.591·37-s − 0.0172·38-s + 0.451·40-s − 1.20·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $-1$
Analytic conductor: \(1.94036\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 243,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.53T + 2T^{2} \)
5 \( 1 + 0.467T + 5T^{2} \)
7 \( 1 + 3.22T + 7T^{2} \)
11 \( 1 - 3.10T + 11T^{2} \)
13 \( 1 + 2.18T + 13T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 0.0418T + 19T^{2} \)
23 \( 1 + 6.10T + 23T^{2} \)
29 \( 1 + 6.57T + 29T^{2} \)
31 \( 1 + 6.22T + 31T^{2} \)
37 \( 1 - 3.59T + 37T^{2} \)
41 \( 1 + 7.70T + 41T^{2} \)
43 \( 1 + 0.588T + 43T^{2} \)
47 \( 1 - 9.66T + 47T^{2} \)
53 \( 1 + 4.95T + 53T^{2} \)
59 \( 1 - 8.53T + 59T^{2} \)
61 \( 1 + 1.26T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 + 8.23T + 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 - 1.50T + 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 - 18.6T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.42460642410637826890522005543, −10.31564173334467064296540622658, −9.541383471047957055748250942967, −8.992831124463235558092316781552, −7.77789329088139499425888908988, −6.91356097673695218347818511037, −6.03587855749369941395695055753, −3.74567657895574150041096517072, −2.08831159546667970910620055785, 0, 2.08831159546667970910620055785, 3.74567657895574150041096517072, 6.03587855749369941395695055753, 6.91356097673695218347818511037, 7.77789329088139499425888908988, 8.992831124463235558092316781552, 9.541383471047957055748250942967, 10.31564173334467064296540622658, 11.42460642410637826890522005543

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