Properties

Label 2-21e2-1.1-c5-0-73
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  + 8.44·2-s + 39.3·4-s − 36·5-s + 61.9·8-s − 304.·10-s − 295.·11-s + 1.14e3·13-s − 735.·16-s + 1.03e3·17-s − 2.10e3·19-s − 1.41e3·20-s − 2.49e3·22-s + 640.·23-s − 1.82e3·25-s + 9.69e3·26-s − 7.63e3·29-s + 966.·31-s − 8.19e3·32-s + 8.71e3·34-s − 1.77e3·37-s − 1.78e4·38-s − 2.23e3·40-s − 1.19e4·41-s − 1.98e4·43-s − 1.16e4·44-s + 5.41e3·46-s − 2.79e4·47-s + ⋯
L(s)  = 1  + 1.49·2-s + 1.22·4-s − 0.643·5-s + 0.342·8-s − 0.961·10-s − 0.736·11-s + 1.88·13-s − 0.718·16-s + 0.866·17-s − 1.33·19-s − 0.791·20-s − 1.09·22-s + 0.252·23-s − 0.585·25-s + 2.81·26-s − 1.68·29-s + 0.180·31-s − 1.41·32-s + 1.29·34-s − 0.212·37-s − 2.00·38-s − 0.220·40-s − 1.11·41-s − 1.63·43-s − 0.905·44-s + 0.377·46-s − 1.84·47-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 \( 1 - 8.44T + 32T^{2} \)
5 \( 1 + 36T + 3.12e3T^{2} \)
11 \( 1 + 295.T + 1.61e5T^{2} \)
13 \( 1 - 1.14e3T + 3.71e5T^{2} \)
17 \( 1 - 1.03e3T + 1.41e6T^{2} \)
19 \( 1 + 2.10e3T + 2.47e6T^{2} \)
23 \( 1 - 640.T + 6.43e6T^{2} \)
29 \( 1 + 7.63e3T + 2.05e7T^{2} \)
31 \( 1 - 966.T + 2.86e7T^{2} \)
37 \( 1 + 1.77e3T + 6.93e7T^{2} \)
41 \( 1 + 1.19e4T + 1.15e8T^{2} \)
43 \( 1 + 1.98e4T + 1.47e8T^{2} \)
47 \( 1 + 2.79e4T + 2.29e8T^{2} \)
53 \( 1 - 7.11e3T + 4.18e8T^{2} \)
59 \( 1 + 2.08e4T + 7.14e8T^{2} \)
61 \( 1 - 2.38e4T + 8.44e8T^{2} \)
67 \( 1 - 3.46e4T + 1.35e9T^{2} \)
71 \( 1 - 2.84e4T + 1.80e9T^{2} \)
73 \( 1 - 1.52e4T + 2.07e9T^{2} \)
79 \( 1 + 7.30e4T + 3.07e9T^{2} \)
83 \( 1 + 3.03e4T + 3.93e9T^{2} \)
89 \( 1 + 3.60e4T + 5.58e9T^{2} \)
97 \( 1 - 1.53e5T + 8.58e9T^{2} \)
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

−10.16796018554225294144428298000, −8.720666654300518761836006854296, −7.958810638465216044128767776331, −6.70410558402122803501774336586, −5.87979325166003528392137598957, −4.98152861759306868455910770502, −3.81958507626244868794461265985, −3.34968875311475324579506409503, −1.81855741798239935593978434442, 0, 1.81855741798239935593978434442, 3.34968875311475324579506409503, 3.81958507626244868794461265985, 4.98152861759306868455910770502, 5.87979325166003528392137598957, 6.70410558402122803501774336586, 7.958810638465216044128767776331, 8.720666654300518761836006854296, 10.16796018554225294144428298000

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