Properties

Degree $2$
Conductor $441$
Sign $1$
Motivic weight $3$
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: $1$
Motivic weight: \(3\)
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(7.867223976\)
\(L(\frac12)\) \(\approx\) \(7.867223976\)
\(L(\frac{5}{2})\) 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

−10.86069240612887171751226808446, −10.14566986301809319416073516147, −9.049435105398708279942348994521, −7.36768464887901727385991048708, −6.52502956480527093864955048262, −5.84126734760816697635337494646, −4.91845599433010237738221723715, −4.01298064682546215002871575029, −2.54341338814675595787958961351, −1.87472276351988175545453895981, 1.87472276351988175545453895981, 2.54341338814675595787958961351, 4.01298064682546215002871575029, 4.91845599433010237738221723715, 5.84126734760816697635337494646, 6.52502956480527093864955048262, 7.36768464887901727385991048708, 9.049435105398708279942348994521, 10.14566986301809319416073516147, 10.86069240612887171751226808446

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