Properties

Label 2-21e2-1.1-c5-0-63
Degree $2$
Conductor $441$
Sign $-1$
Analytic cond. $70.7292$
Root an. cond. $8.41006$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.72·2-s − 29.0·4-s + 73.1·5-s + 105.·8-s − 125.·10-s − 193.·11-s + 151.·13-s + 747.·16-s − 1.76e3·17-s + 1.12e3·19-s − 2.12e3·20-s + 332.·22-s − 2.41e3·23-s + 2.22e3·25-s − 261.·26-s + 7.72e3·29-s − 8.17e3·31-s − 4.65e3·32-s + 3.03e3·34-s + 5.93e3·37-s − 1.94e3·38-s + 7.68e3·40-s − 4.97e3·41-s − 5.65e3·43-s + 5.60e3·44-s + 4.15e3·46-s − 4.72e3·47-s + ⋯
L(s)  = 1  − 0.304·2-s − 0.907·4-s + 1.30·5-s + 0.580·8-s − 0.398·10-s − 0.481·11-s + 0.248·13-s + 0.730·16-s − 1.47·17-s + 0.716·19-s − 1.18·20-s + 0.146·22-s − 0.951·23-s + 0.711·25-s − 0.0757·26-s + 1.70·29-s − 1.52·31-s − 0.803·32-s + 0.449·34-s + 0.712·37-s − 0.218·38-s + 0.759·40-s − 0.462·41-s − 0.466·43-s + 0.436·44-s + 0.289·46-s − 0.311·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(70.7292\)
Root analytic conductor: \(8.41006\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 441,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 1.72T + 32T^{2} \)
5 \( 1 - 73.1T + 3.12e3T^{2} \)
11 \( 1 + 193.T + 1.61e5T^{2} \)
13 \( 1 - 151.T + 3.71e5T^{2} \)
17 \( 1 + 1.76e3T + 1.41e6T^{2} \)
19 \( 1 - 1.12e3T + 2.47e6T^{2} \)
23 \( 1 + 2.41e3T + 6.43e6T^{2} \)
29 \( 1 - 7.72e3T + 2.05e7T^{2} \)
31 \( 1 + 8.17e3T + 2.86e7T^{2} \)
37 \( 1 - 5.93e3T + 6.93e7T^{2} \)
41 \( 1 + 4.97e3T + 1.15e8T^{2} \)
43 \( 1 + 5.65e3T + 1.47e8T^{2} \)
47 \( 1 + 4.72e3T + 2.29e8T^{2} \)
53 \( 1 - 3.20e4T + 4.18e8T^{2} \)
59 \( 1 - 2.85e4T + 7.14e8T^{2} \)
61 \( 1 - 2.90e4T + 8.44e8T^{2} \)
67 \( 1 + 3.05e4T + 1.35e9T^{2} \)
71 \( 1 + 5.69e4T + 1.80e9T^{2} \)
73 \( 1 - 9.72e3T + 2.07e9T^{2} \)
79 \( 1 + 9.84e4T + 3.07e9T^{2} \)
83 \( 1 + 5.06e4T + 3.93e9T^{2} \)
89 \( 1 + 5.96e4T + 5.58e9T^{2} \)
97 \( 1 + 1.34e5T + 8.58e9T^{2} \)
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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

−9.914512127507345336400521554630, −8.987727541843118663172708516952, −8.351100452497072129242880882527, −7.07698100560862656146132187989, −5.94204535024031199336278227731, −5.14871096353512985023412379024, −4.08208364268789818599596325033, −2.54083322087813541063759708801, −1.39182401357811451403857198434, 0, 1.39182401357811451403857198434, 2.54083322087813541063759708801, 4.08208364268789818599596325033, 5.14871096353512985023412379024, 5.94204535024031199336278227731, 7.07698100560862656146132187989, 8.351100452497072129242880882527, 8.987727541843118663172708516952, 9.914512127507345336400521554630

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