Properties

Label 2-21e2-1.1-c1-0-0
Degree $2$
Conductor $441$
Sign $1$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
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 − 3.46·5-s + 1.73·8-s + 5.99·10-s − 3.46·11-s − 2·13-s − 5·16-s + 3.46·17-s + 4·19-s − 3.46·20-s + 5.99·22-s + 3.46·23-s + 6.99·25-s + 3.46·26-s + 4·31-s + 5.19·32-s − 5.99·34-s + 2·37-s − 6.92·38-s − 6.00·40-s + 10.3·41-s − 4·43-s − 3.46·44-s − 5.99·46-s + 6.92·47-s − 12.1·50-s + ⋯
L(s)  = 1  − 1.22·2-s + 0.499·4-s − 1.54·5-s + 0.612·8-s + 1.89·10-s − 1.04·11-s − 0.554·13-s − 1.25·16-s + 0.840·17-s + 0.917·19-s − 0.774·20-s + 1.27·22-s + 0.722·23-s + 1.39·25-s + 0.679·26-s + 0.718·31-s + 0.918·32-s − 1.02·34-s + 0.328·37-s − 1.12·38-s − 0.948·40-s + 1.62·41-s − 0.609·43-s − 0.522·44-s − 0.884·46-s + 1.01·47-s − 1.71·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4408106826\)
\(L(\frac12)\) \(\approx\) \(0.4408106826\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + 1.73T + 2T^{2} \)
5 \( 1 + 3.46T + 5T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 - 6.92T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−10.95658054654902885523846772136, −10.17188386146879038957946866480, −9.268646300567467243411926807400, −8.225844865232840812413709801409, −7.69810934092326468321480626486, −7.11368080298575350370680898060, −5.29179036278425807274949680025, −4.23077073410533406981583562252, −2.85720623146223462659941831225, −0.74095505323005833418238937755, 0.74095505323005833418238937755, 2.85720623146223462659941831225, 4.23077073410533406981583562252, 5.29179036278425807274949680025, 7.11368080298575350370680898060, 7.69810934092326468321480626486, 8.225844865232840812413709801409, 9.268646300567467243411926807400, 10.17188386146879038957946866480, 10.95658054654902885523846772136

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