Properties

Label 2-3e5-3.2-c0-0-0
Degree $2$
Conductor $243$
Sign $1$
Analytic cond. $0.121272$
Root an. cond. $0.348242$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 7-s − 13-s + 16-s − 19-s + 25-s − 28-s − 31-s − 37-s − 43-s − 52-s + 2·61-s + 64-s + 2·67-s + 2·73-s − 76-s − 79-s + 91-s − 97-s + 100-s + 2·103-s − 109-s − 112-s + ⋯
L(s)  = 1  + 4-s − 7-s − 13-s + 16-s − 19-s + 25-s − 28-s − 31-s − 37-s − 43-s − 52-s + 2·61-s + 64-s + 2·67-s + 2·73-s − 76-s − 79-s + 91-s − 97-s + 100-s + 2·103-s − 109-s − 112-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $1$
Analytic conductor: \(0.121272\)
Root analytic conductor: \(0.348242\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{243} (242, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 243,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7972813154\)
\(L(\frac12)\) \(\approx\) \(0.7972813154\)
\(L(1)\) 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 \( ( 1 - T )( 1 + T ) \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )^{2} \)
67 \( ( 1 - T )^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )^{2} \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T + T^{2} \)
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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

−12.46692594800999402144650086625, −11.37038200707344258992550771187, −10.43871971974208712818352306125, −9.644675198453474443560245231097, −8.387781095284297023473603737410, −7.09247574894837359739474398716, −6.53938700992242303793050246921, −5.24416125057537026440986145054, −3.52875960380466377559775254870, −2.29360994373471085779290231288, 2.29360994373471085779290231288, 3.52875960380466377559775254870, 5.24416125057537026440986145054, 6.53938700992242303793050246921, 7.09247574894837359739474398716, 8.387781095284297023473603737410, 9.644675198453474443560245231097, 10.43871971974208712818352306125, 11.37038200707344258992550771187, 12.46692594800999402144650086625

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