Properties

Label 2-21e2-1.1-c5-0-7
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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.81·2-s − 24.0·4-s − 45.9·5-s + 157.·8-s + 129.·10-s + 551.·11-s − 1.09e3·13-s + 326.·16-s − 1.18e3·17-s + 1.16e3·19-s + 1.10e3·20-s − 1.55e3·22-s − 44.3·23-s − 1.00e3·25-s + 3.07e3·26-s − 3.32e3·29-s − 8.78e3·31-s − 5.96e3·32-s + 3.32e3·34-s − 2.55e3·37-s − 3.28e3·38-s − 7.25e3·40-s − 1.27e4·41-s − 96.7·43-s − 1.32e4·44-s + 124.·46-s + 7.67e3·47-s + ⋯
L(s)  = 1  − 0.497·2-s − 0.752·4-s − 0.822·5-s + 0.872·8-s + 0.409·10-s + 1.37·11-s − 1.79·13-s + 0.318·16-s − 0.990·17-s + 0.741·19-s + 0.618·20-s − 0.684·22-s − 0.0174·23-s − 0.323·25-s + 0.893·26-s − 0.735·29-s − 1.64·31-s − 1.03·32-s + 0.493·34-s − 0.307·37-s − 0.368·38-s − 0.717·40-s − 1.18·41-s − 0.00798·43-s − 1.03·44-s + 0.00870·46-s + 0.507·47-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5496734381\)
\(L(\frac12)\) \(\approx\) \(0.5496734381\)
\(L(\frac{7}{2})\) 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 \( 1 + 2.81T + 32T^{2} \)
5 \( 1 + 45.9T + 3.12e3T^{2} \)
11 \( 1 - 551.T + 1.61e5T^{2} \)
13 \( 1 + 1.09e3T + 3.71e5T^{2} \)
17 \( 1 + 1.18e3T + 1.41e6T^{2} \)
19 \( 1 - 1.16e3T + 2.47e6T^{2} \)
23 \( 1 + 44.3T + 6.43e6T^{2} \)
29 \( 1 + 3.32e3T + 2.05e7T^{2} \)
31 \( 1 + 8.78e3T + 2.86e7T^{2} \)
37 \( 1 + 2.55e3T + 6.93e7T^{2} \)
41 \( 1 + 1.27e4T + 1.15e8T^{2} \)
43 \( 1 + 96.7T + 1.47e8T^{2} \)
47 \( 1 - 7.67e3T + 2.29e8T^{2} \)
53 \( 1 - 1.19e4T + 4.18e8T^{2} \)
59 \( 1 - 9.85e3T + 7.14e8T^{2} \)
61 \( 1 - 3.85e4T + 8.44e8T^{2} \)
67 \( 1 + 6.75e4T + 1.35e9T^{2} \)
71 \( 1 - 6.13e4T + 1.80e9T^{2} \)
73 \( 1 + 1.85e3T + 2.07e9T^{2} \)
79 \( 1 + 8.52T + 3.07e9T^{2} \)
83 \( 1 + 9.50e4T + 3.93e9T^{2} \)
89 \( 1 - 5.36e4T + 5.58e9T^{2} \)
97 \( 1 + 3.11e3T + 8.58e9T^{2} \)
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

−10.05828321790998462488948456395, −9.338773922007920851910631474323, −8.673631018755915342642276049489, −7.53311820640819551093901377005, −7.01095288841720600085506661374, −5.38496528986627373950642193570, −4.39643888911104012084316368324, −3.60585309500521017549745403225, −1.86845692562583794574536132190, −0.41282025781609969284278534725, 0.41282025781609969284278534725, 1.86845692562583794574536132190, 3.60585309500521017549745403225, 4.39643888911104012084316368324, 5.38496528986627373950642193570, 7.01095288841720600085506661374, 7.53311820640819551093901377005, 8.673631018755915342642276049489, 9.338773922007920851910631474323, 10.05828321790998462488948456395

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