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Properties

Label 2-21e2-21.20-c3-0-29
Degree $2$
Conductor $441$
Sign $-0.716 + 0.698i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.65i·2-s − 5.34·4-s + 17.3·5-s − 9.68i·8-s − 63.5i·10-s − 32.5i·11-s + 42.4i·13-s − 78.1·16-s + 85.6·17-s − 69.1i·19-s − 93.0·20-s − 118.·22-s + 167. i·23-s + 177.·25-s + 155.·26-s + ⋯
L(s)  = 1  − 1.29i·2-s − 0.668·4-s + 1.55·5-s − 0.428i·8-s − 2.01i·10-s − 0.892i·11-s + 0.906i·13-s − 1.22·16-s + 1.22·17-s − 0.835i·19-s − 1.04·20-s − 1.15·22-s + 1.51i·23-s + 1.42·25-s + 1.17·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 + 0.698i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.716 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.716 + 0.698i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.716 + 0.698i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.693334214\)
\(L(\frac12)\) \(\approx\) \(2.693334214\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.65iT - 8T^{2} \)
5 \( 1 - 17.3T + 125T^{2} \)
11 \( 1 + 32.5iT - 1.33e3T^{2} \)
13 \( 1 - 42.4iT - 2.19e3T^{2} \)
17 \( 1 - 85.6T + 4.91e3T^{2} \)
19 \( 1 + 69.1iT - 6.85e3T^{2} \)
23 \( 1 - 167. iT - 1.21e4T^{2} \)
29 \( 1 + 254. iT - 2.43e4T^{2} \)
31 \( 1 + 325. iT - 2.97e4T^{2} \)
37 \( 1 - 345.T + 5.06e4T^{2} \)
41 \( 1 + 182.T + 6.89e4T^{2} \)
43 \( 1 - 140.T + 7.95e4T^{2} \)
47 \( 1 + 43.0T + 1.03e5T^{2} \)
53 \( 1 - 193. iT - 1.48e5T^{2} \)
59 \( 1 - 111.T + 2.05e5T^{2} \)
61 \( 1 + 90.5iT - 2.26e5T^{2} \)
67 \( 1 + 1.04e3T + 3.00e5T^{2} \)
71 \( 1 - 464. iT - 3.57e5T^{2} \)
73 \( 1 + 182. iT - 3.89e5T^{2} \)
79 \( 1 + 238.T + 4.93e5T^{2} \)
83 \( 1 + 375.T + 5.71e5T^{2} \)
89 \( 1 - 1.43e3T + 7.04e5T^{2} \)
97 \( 1 + 638. iT - 9.12e5T^{2} \)
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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

−10.32329381322111151260053751290, −9.558836905076642240057354854431, −9.234665206846873920319736259399, −7.68007939658459035128211330173, −6.31383038451233965909753945768, −5.63872029228510417776913803588, −4.18414480754946804421916118343, −2.92235913936343662402749319884, −2.00669814473071578504537038149, −0.919983241026501020906623959606, 1.53442558767175813544066663610, 2.87226349089705445010242575646, 4.85478008686433079434302365149, 5.55144073141553076312456264242, 6.32946166853737241947995533663, 7.17862418107245146613331320626, 8.176847219539132354504266095847, 9.074671032704934835485390147303, 10.11551094258772432874688776909, 10.58763368650732520989603552157

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