Properties

Label 2-3e5-1.1-c1-0-0
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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·2-s + 0.999·4-s − 3.46·5-s − 7-s + 1.73·8-s + 5.99·10-s + 3.46·11-s + 5·13-s + 1.73·14-s − 5·16-s − 19-s − 3.46·20-s − 5.99·22-s + 6.92·23-s + 6.99·25-s − 8.66·26-s − 0.999·28-s + 3.46·29-s + 5·31-s + 5.19·32-s + 3.46·35-s − 37-s + 1.73·38-s − 6.00·40-s − 3.46·41-s − 43-s + 3.46·44-s + ⋯
L(s)  = 1  − 1.22·2-s + 0.499·4-s − 1.54·5-s − 0.377·7-s + 0.612·8-s + 1.89·10-s + 1.04·11-s + 1.38·13-s + 0.462·14-s − 1.25·16-s − 0.229·19-s − 0.774·20-s − 1.27·22-s + 1.44·23-s + 1.39·25-s − 1.69·26-s − 0.188·28-s + 0.643·29-s + 0.898·31-s + 0.918·32-s + 0.585·35-s − 0.164·37-s + 0.280·38-s − 0.948·40-s − 0.541·41-s − 0.152·43-s + 0.522·44-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: \(0\)
Selberg data: \((2,\ 243,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5108811642\)
\(L(\frac12)\) \(\approx\) \(0.5108811642\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73T + 2T^{2} \)
5 \( 1 + 3.46T + 5T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 - 6.92T + 23T^{2} \)
29 \( 1 - 3.46T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 + 6.92T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 - 17T + 97T^{2} \)
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

−11.68613044632300397194764208501, −11.14495484165665719309580933705, −10.13654896803195556492580659962, −8.833245131673608769163171550514, −8.519652151235185896572998162213, −7.37225555795780709109469246797, −6.51667717431085117725053286253, −4.50551446413152782472606775083, −3.47172281941875945571553746616, −0.970090806857385628640518965887, 0.970090806857385628640518965887, 3.47172281941875945571553746616, 4.50551446413152782472606775083, 6.51667717431085117725053286253, 7.37225555795780709109469246797, 8.519652151235185896572998162213, 8.833245131673608769163171550514, 10.13654896803195556492580659962, 11.14495484165665719309580933705, 11.68613044632300397194764208501

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