Properties

Label 2-21e2-1.1-c3-0-45
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  + 4.05·2-s + 8.44·4-s − 9.92·5-s + 1.80·8-s − 40.2·10-s + 13.5·11-s − 18.5·13-s − 60.2·16-s − 93.7·17-s − 131.·19-s − 83.7·20-s + 54.9·22-s + 198.·23-s − 26.5·25-s − 75.2·26-s − 188.·29-s + 83.9·31-s − 258.·32-s − 380.·34-s + 80.1·37-s − 534.·38-s − 17.9·40-s − 385.·41-s − 397.·43-s + 114.·44-s + 804.·46-s + 272.·47-s + ⋯
L(s)  = 1  + 1.43·2-s + 1.05·4-s − 0.887·5-s + 0.0799·8-s − 1.27·10-s + 0.371·11-s − 0.395·13-s − 0.941·16-s − 1.33·17-s − 1.59·19-s − 0.936·20-s + 0.532·22-s + 1.79·23-s − 0.212·25-s − 0.567·26-s − 1.20·29-s + 0.486·31-s − 1.42·32-s − 1.91·34-s + 0.356·37-s − 2.28·38-s − 0.0709·40-s − 1.46·41-s − 1.40·43-s + 0.391·44-s + 2.57·46-s + 0.845·47-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 \( 1 - 4.05T + 8T^{2} \)
5 \( 1 + 9.92T + 125T^{2} \)
11 \( 1 - 13.5T + 1.33e3T^{2} \)
13 \( 1 + 18.5T + 2.19e3T^{2} \)
17 \( 1 + 93.7T + 4.91e3T^{2} \)
19 \( 1 + 131.T + 6.85e3T^{2} \)
23 \( 1 - 198.T + 1.21e4T^{2} \)
29 \( 1 + 188.T + 2.43e4T^{2} \)
31 \( 1 - 83.9T + 2.97e4T^{2} \)
37 \( 1 - 80.1T + 5.06e4T^{2} \)
41 \( 1 + 385.T + 6.89e4T^{2} \)
43 \( 1 + 397.T + 7.95e4T^{2} \)
47 \( 1 - 272.T + 1.03e5T^{2} \)
53 \( 1 - 36.9T + 1.48e5T^{2} \)
59 \( 1 - 395.T + 2.05e5T^{2} \)
61 \( 1 + 13.4T + 2.26e5T^{2} \)
67 \( 1 - 340.T + 3.00e5T^{2} \)
71 \( 1 - 211.T + 3.57e5T^{2} \)
73 \( 1 + 486.T + 3.89e5T^{2} \)
79 \( 1 - 293.T + 4.93e5T^{2} \)
83 \( 1 - 889.T + 5.71e5T^{2} \)
89 \( 1 - 1.14e3T + 7.04e5T^{2} \)
97 \( 1 + 1.38e3T + 9.12e5T^{2} \)
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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

−10.71731687995324917678335132065, −9.218704293967087810490680703712, −8.401594841150004334637770900432, −7.05502429206257969010356626111, −6.44773557013951570054889898880, −5.12425670975547098620064535641, −4.34021411218410451899138453955, −3.53252452496670269995780672539, −2.25400953870055639832462913043, 0, 2.25400953870055639832462913043, 3.53252452496670269995780672539, 4.34021411218410451899138453955, 5.12425670975547098620064535641, 6.44773557013951570054889898880, 7.05502429206257969010356626111, 8.401594841150004334637770900432, 9.218704293967087810490680703712, 10.71731687995324917678335132065

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