Properties

Label 2-21e2-1.1-c5-0-58
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6.38·2-s + 8.83·4-s + 38.7·5-s + 148.·8-s − 247.·10-s + 576.·11-s − 391.·13-s − 1.22e3·16-s − 1.32e3·17-s − 942.·19-s + 341.·20-s − 3.68e3·22-s + 1.63e3·23-s − 1.62e3·25-s + 2.50e3·26-s + 1.46e3·29-s + 3.91e3·31-s + 3.11e3·32-s + 8.49e3·34-s − 1.63e4·37-s + 6.02e3·38-s + 5.73e3·40-s − 1.31e4·41-s + 1.47e4·43-s + 5.08e3·44-s − 1.04e4·46-s − 6.81e3·47-s + ⋯
L(s)  = 1  − 1.12·2-s + 0.275·4-s + 0.692·5-s + 0.817·8-s − 0.782·10-s + 1.43·11-s − 0.642·13-s − 1.19·16-s − 1.11·17-s − 0.598·19-s + 0.191·20-s − 1.62·22-s + 0.643·23-s − 0.520·25-s + 0.725·26-s + 0.323·29-s + 0.731·31-s + 0.537·32-s + 1.26·34-s − 1.95·37-s + 0.676·38-s + 0.566·40-s − 1.21·41-s + 1.21·43-s + 0.396·44-s − 0.726·46-s − 0.449·47-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 \( 1 + 6.38T + 32T^{2} \)
5 \( 1 - 38.7T + 3.12e3T^{2} \)
11 \( 1 - 576.T + 1.61e5T^{2} \)
13 \( 1 + 391.T + 3.71e5T^{2} \)
17 \( 1 + 1.32e3T + 1.41e6T^{2} \)
19 \( 1 + 942.T + 2.47e6T^{2} \)
23 \( 1 - 1.63e3T + 6.43e6T^{2} \)
29 \( 1 - 1.46e3T + 2.05e7T^{2} \)
31 \( 1 - 3.91e3T + 2.86e7T^{2} \)
37 \( 1 + 1.63e4T + 6.93e7T^{2} \)
41 \( 1 + 1.31e4T + 1.15e8T^{2} \)
43 \( 1 - 1.47e4T + 1.47e8T^{2} \)
47 \( 1 + 6.81e3T + 2.29e8T^{2} \)
53 \( 1 - 2.01e3T + 4.18e8T^{2} \)
59 \( 1 - 5.14e4T + 7.14e8T^{2} \)
61 \( 1 + 4.10e4T + 8.44e8T^{2} \)
67 \( 1 - 5.05e4T + 1.35e9T^{2} \)
71 \( 1 + 3.99e4T + 1.80e9T^{2} \)
73 \( 1 - 5.56e4T + 2.07e9T^{2} \)
79 \( 1 + 6.31e4T + 3.07e9T^{2} \)
83 \( 1 - 4.55e4T + 3.93e9T^{2} \)
89 \( 1 - 1.56e4T + 5.58e9T^{2} \)
97 \( 1 + 3.12e3T + 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.712000892620380072860334888702, −9.012427557239995462750680318216, −8.399402264299868026734800389322, −7.07295666054133509186864911561, −6.47697299208237884606306989848, −5.03968112962672072890145470059, −3.99043836664774357386062073019, −2.24669176330310343385865794890, −1.31064231601552648464275750368, 0, 1.31064231601552648464275750368, 2.24669176330310343385865794890, 3.99043836664774357386062073019, 5.03968112962672072890145470059, 6.47697299208237884606306989848, 7.07295666054133509186864911561, 8.399402264299868026734800389322, 9.012427557239995462750680318216, 9.712000892620380072860334888702

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