Properties

Label 2-21e2-1.1-c3-0-0
Degree $2$
Conductor $441$
Sign $1$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.39·2-s + 21.1·4-s − 15.5·5-s − 71.0·8-s + 84.1·10-s − 31.9·11-s − 72.5·13-s + 214.·16-s + 29.0·17-s − 108.·19-s − 329.·20-s + 172.·22-s + 55.2·23-s + 117.·25-s + 391.·26-s − 17.7·29-s − 56.1·31-s − 589.·32-s − 156.·34-s − 295.·37-s + 587.·38-s + 1.10e3·40-s − 238.·41-s + 16.8·43-s − 676.·44-s − 298.·46-s − 511.·47-s + ⋯
L(s)  = 1  − 1.90·2-s + 2.64·4-s − 1.39·5-s − 3.14·8-s + 2.65·10-s − 0.876·11-s − 1.54·13-s + 3.35·16-s + 0.414·17-s − 1.31·19-s − 3.68·20-s + 1.67·22-s + 0.501·23-s + 0.941·25-s + 2.95·26-s − 0.113·29-s − 0.325·31-s − 3.25·32-s − 0.790·34-s − 1.31·37-s + 2.50·38-s + 4.37·40-s − 0.908·41-s + 0.0596·43-s − 2.31·44-s − 0.956·46-s − 1.58·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1517023326\)
\(L(\frac12)\) \(\approx\) \(0.1517023326\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.39T + 8T^{2} \)
5 \( 1 + 15.5T + 125T^{2} \)
11 \( 1 + 31.9T + 1.33e3T^{2} \)
13 \( 1 + 72.5T + 2.19e3T^{2} \)
17 \( 1 - 29.0T + 4.91e3T^{2} \)
19 \( 1 + 108.T + 6.85e3T^{2} \)
23 \( 1 - 55.2T + 1.21e4T^{2} \)
29 \( 1 + 17.7T + 2.43e4T^{2} \)
31 \( 1 + 56.1T + 2.97e4T^{2} \)
37 \( 1 + 295.T + 5.06e4T^{2} \)
41 \( 1 + 238.T + 6.89e4T^{2} \)
43 \( 1 - 16.8T + 7.95e4T^{2} \)
47 \( 1 + 511.T + 1.03e5T^{2} \)
53 \( 1 + 265.T + 1.48e5T^{2} \)
59 \( 1 - 254.T + 2.05e5T^{2} \)
61 \( 1 - 72.8T + 2.26e5T^{2} \)
67 \( 1 + 506.T + 3.00e5T^{2} \)
71 \( 1 - 827.T + 3.57e5T^{2} \)
73 \( 1 - 372.T + 3.89e5T^{2} \)
79 \( 1 - 1.02e3T + 4.93e5T^{2} \)
83 \( 1 + 453.T + 5.71e5T^{2} \)
89 \( 1 + 332.T + 7.04e5T^{2} \)
97 \( 1 - 1.16e3T + 9.12e5T^{2} \)
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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

−10.52952190155467216639937612072, −9.823589200248009720371697844744, −8.733004002651335090085062258055, −8.018645691964753225436586557837, −7.43536090154121925700151175931, −6.65143078347646140069836865049, −5.00086308435493138611326150767, −3.29723169708872725086202042218, −2.09357561229777712202459988683, −0.30800214825876499867451036834, 0.30800214825876499867451036834, 2.09357561229777712202459988683, 3.29723169708872725086202042218, 5.00086308435493138611326150767, 6.65143078347646140069836865049, 7.43536090154121925700151175931, 8.018645691964753225436586557837, 8.733004002651335090085062258055, 9.823589200248009720371697844744, 10.52952190155467216639937612072

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