Properties

Label 2-21e2-1.1-c1-0-4
Degree $2$
Conductor $441$
Sign $1$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 2·5-s − 4·10-s + 2·11-s + 13-s − 4·16-s + 19-s + 4·20-s − 4·22-s − 25-s − 2·26-s − 4·29-s + 9·31-s + 8·32-s + 3·37-s − 2·38-s + 10·41-s + 5·43-s + 4·44-s + 6·47-s + 2·50-s + 2·52-s − 12·53-s + 4·55-s + 8·58-s + 12·59-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.894·5-s − 1.26·10-s + 0.603·11-s + 0.277·13-s − 16-s + 0.229·19-s + 0.894·20-s − 0.852·22-s − 1/5·25-s − 0.392·26-s − 0.742·29-s + 1.61·31-s + 1.41·32-s + 0.493·37-s − 0.324·38-s + 1.56·41-s + 0.762·43-s + 0.603·44-s + 0.875·47-s + 0.282·50-s + 0.277·52-s − 1.64·53-s + 0.539·55-s + 1.05·58-s + 1.56·59-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8379801548\)
\(L(\frac12)\) \(\approx\) \(0.8379801548\)
\(L(\frac{3}{2})\) 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 \( 1 + p T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−10.87139152298459419633900996880, −9.879400023159387748546449740681, −9.459031124333216147077448358277, −8.583706845901838358159961836152, −7.67312246287158311998952783143, −6.65130832285966559117996014878, −5.72037257717675019727399870073, −4.23783192793666623809357478808, −2.43327032489084752763662196914, −1.15383453307155328641933378041, 1.15383453307155328641933378041, 2.43327032489084752763662196914, 4.23783192793666623809357478808, 5.72037257717675019727399870073, 6.65130832285966559117996014878, 7.67312246287158311998952783143, 8.583706845901838358159961836152, 9.459031124333216147077448358277, 9.879400023159387748546449740681, 10.87139152298459419633900996880

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