Properties

Label 2-3e4-1.1-c1-0-1
Degree $2$
Conductor $81$
Sign $1$
Analytic cond. $0.646788$
Root an. cond. $0.804231$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·2-s + 0.999·4-s − 1.73·5-s + 2·7-s − 1.73·8-s − 2.99·10-s − 3.46·11-s − 13-s + 3.46·14-s − 5·16-s + 5.19·17-s + 2·19-s − 1.73·20-s − 5.99·22-s + 3.46·23-s − 2.00·25-s − 1.73·26-s + 1.99·28-s + 1.73·29-s + 8·31-s − 5.19·32-s + 9·34-s − 3.46·35-s − 7·37-s + 3.46·38-s + 3.00·40-s − 6.92·41-s + ⋯
L(s)  = 1  + 1.22·2-s + 0.499·4-s − 0.774·5-s + 0.755·7-s − 0.612·8-s − 0.948·10-s − 1.04·11-s − 0.277·13-s + 0.925·14-s − 1.25·16-s + 1.26·17-s + 0.458·19-s − 0.387·20-s − 1.27·22-s + 0.722·23-s − 0.400·25-s − 0.339·26-s + 0.377·28-s + 0.321·29-s + 1.43·31-s − 0.918·32-s + 1.54·34-s − 0.585·35-s − 1.15·37-s + 0.561·38-s + 0.474·40-s − 1.08·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $1$
Analytic conductor: \(0.646788\)
Root analytic conductor: \(0.804231\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.435104167\)
\(L(\frac12)\) \(\approx\) \(1.435104167\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73T + 5T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 - 5.19T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 - 1.73T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + 5.19T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−14.29964332369256398717487679596, −13.39369631200027704092706736493, −12.22520072356073210718671997465, −11.60281525157102031033473031367, −10.20242410709674926896312591641, −8.462010452150115627851294225307, −7.36311860625225098919703816696, −5.56324876902780652194785160059, −4.60289706621031782825417838661, −3.12234009529020877746547298161, 3.12234009529020877746547298161, 4.60289706621031782825417838661, 5.56324876902780652194785160059, 7.36311860625225098919703816696, 8.462010452150115627851294225307, 10.20242410709674926896312591641, 11.60281525157102031033473031367, 12.22520072356073210718671997465, 13.39369631200027704092706736493, 14.29964332369256398717487679596

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