Properties

Label 2-21e2-1.1-c3-0-38
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.248·2-s − 7.93·4-s + 12.4·5-s + 3.95·8-s − 3.08·10-s − 60.3·11-s + 36.4·13-s + 62.5·16-s + 48.7·17-s − 50.5·19-s − 98.7·20-s + 14.9·22-s − 138.·23-s + 29.6·25-s − 9.03·26-s + 61.1·29-s − 1.16·31-s − 47.1·32-s − 12.0·34-s + 69.5·37-s + 12.5·38-s + 49.1·40-s − 308.·41-s + 174.·43-s + 478.·44-s + 34.4·46-s − 389.·47-s + ⋯
L(s)  = 1  − 0.0877·2-s − 0.992·4-s + 1.11·5-s + 0.174·8-s − 0.0975·10-s − 1.65·11-s + 0.777·13-s + 0.976·16-s + 0.695·17-s − 0.610·19-s − 1.10·20-s + 0.144·22-s − 1.25·23-s + 0.236·25-s − 0.0681·26-s + 0.391·29-s − 0.00677·31-s − 0.260·32-s − 0.0609·34-s + 0.308·37-s + 0.0535·38-s + 0.194·40-s − 1.17·41-s + 0.618·43-s + 1.64·44-s + 0.110·46-s − 1.20·47-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: \(1\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 0.248T + 8T^{2} \)
5 \( 1 - 12.4T + 125T^{2} \)
11 \( 1 + 60.3T + 1.33e3T^{2} \)
13 \( 1 - 36.4T + 2.19e3T^{2} \)
17 \( 1 - 48.7T + 4.91e3T^{2} \)
19 \( 1 + 50.5T + 6.85e3T^{2} \)
23 \( 1 + 138.T + 1.21e4T^{2} \)
29 \( 1 - 61.1T + 2.43e4T^{2} \)
31 \( 1 + 1.16T + 2.97e4T^{2} \)
37 \( 1 - 69.5T + 5.06e4T^{2} \)
41 \( 1 + 308.T + 6.89e4T^{2} \)
43 \( 1 - 174.T + 7.95e4T^{2} \)
47 \( 1 + 389.T + 1.03e5T^{2} \)
53 \( 1 + 314.T + 1.48e5T^{2} \)
59 \( 1 + 844.T + 2.05e5T^{2} \)
61 \( 1 + 338.T + 2.26e5T^{2} \)
67 \( 1 + 971.T + 3.00e5T^{2} \)
71 \( 1 - 98.4T + 3.57e5T^{2} \)
73 \( 1 - 710.T + 3.89e5T^{2} \)
79 \( 1 + 486.T + 4.93e5T^{2} \)
83 \( 1 + 605.T + 5.71e5T^{2} \)
89 \( 1 + 218.T + 7.04e5T^{2} \)
97 \( 1 + 782.T + 9.12e5T^{2} \)
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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

−10.15328840944638458773017871456, −9.503430972691324840678557170541, −8.409214981297927113919544277268, −7.79957486622522184973917061394, −6.16661235251940828925483697734, −5.50989562642621202121009616004, −4.51847962085970255774024643158, −3.10327213086883755845245355050, −1.67656957342707491011439612642, 0, 1.67656957342707491011439612642, 3.10327213086883755845245355050, 4.51847962085970255774024643158, 5.50989562642621202121009616004, 6.16661235251940828925483697734, 7.79957486622522184973917061394, 8.409214981297927113919544277268, 9.503430972691324840678557170541, 10.15328840944638458773017871456

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