Properties

Label 2-21e2-1.1-c5-0-6
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  − 2.74·2-s − 24.4·4-s − 58.3·5-s + 155.·8-s + 160.·10-s − 17.4·11-s + 889.·13-s + 356.·16-s − 1.02e3·17-s − 1.73e3·19-s + 1.42e3·20-s + 47.8·22-s − 3.93e3·23-s + 281.·25-s − 2.44e3·26-s − 5.63e3·29-s − 3.09e3·31-s − 5.94e3·32-s + 2.82e3·34-s + 5.02e3·37-s + 4.78e3·38-s − 9.05e3·40-s − 1.83e4·41-s − 1.63e3·43-s + 425.·44-s + 1.08e4·46-s + 9.60e3·47-s + ⋯
L(s)  = 1  − 0.485·2-s − 0.763·4-s − 1.04·5-s + 0.856·8-s + 0.507·10-s − 0.0434·11-s + 1.46·13-s + 0.347·16-s − 0.861·17-s − 1.10·19-s + 0.797·20-s + 0.0210·22-s − 1.55·23-s + 0.0901·25-s − 0.709·26-s − 1.24·29-s − 0.578·31-s − 1.02·32-s + 0.418·34-s + 0.603·37-s + 0.537·38-s − 0.894·40-s − 1.70·41-s − 0.134·43-s + 0.0331·44-s + 0.753·46-s + 0.634·47-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

\(L(3)\) \(\approx\) \(0.5009835237\)
\(L(\frac12)\) \(\approx\) \(0.5009835237\)
\(L(\frac{7}{2})\) 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 \( 1 + 2.74T + 32T^{2} \)
5 \( 1 + 58.3T + 3.12e3T^{2} \)
11 \( 1 + 17.4T + 1.61e5T^{2} \)
13 \( 1 - 889.T + 3.71e5T^{2} \)
17 \( 1 + 1.02e3T + 1.41e6T^{2} \)
19 \( 1 + 1.73e3T + 2.47e6T^{2} \)
23 \( 1 + 3.93e3T + 6.43e6T^{2} \)
29 \( 1 + 5.63e3T + 2.05e7T^{2} \)
31 \( 1 + 3.09e3T + 2.86e7T^{2} \)
37 \( 1 - 5.02e3T + 6.93e7T^{2} \)
41 \( 1 + 1.83e4T + 1.15e8T^{2} \)
43 \( 1 + 1.63e3T + 1.47e8T^{2} \)
47 \( 1 - 9.60e3T + 2.29e8T^{2} \)
53 \( 1 - 2.32e4T + 4.18e8T^{2} \)
59 \( 1 - 3.60e3T + 7.14e8T^{2} \)
61 \( 1 - 2.28e4T + 8.44e8T^{2} \)
67 \( 1 - 4.70e4T + 1.35e9T^{2} \)
71 \( 1 - 1.59e3T + 1.80e9T^{2} \)
73 \( 1 - 5.93e3T + 2.07e9T^{2} \)
79 \( 1 + 8.84e4T + 3.07e9T^{2} \)
83 \( 1 - 9.58e4T + 3.93e9T^{2} \)
89 \( 1 - 4.65e4T + 5.58e9T^{2} \)
97 \( 1 + 7.59e4T + 8.58e9T^{2} \)
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

−10.35698240395337473760883098579, −9.220805991549512824169296684717, −8.417998782571534870749689347403, −7.949886587238886812849391663884, −6.73933817755199891664261293165, −5.54945822207082317598282328618, −4.11142540964067218662149398226, −3.82535613665763969579904239496, −1.84714680412975526335315305331, −0.39480709810407028211308373607, 0.39480709810407028211308373607, 1.84714680412975526335315305331, 3.82535613665763969579904239496, 4.11142540964067218662149398226, 5.54945822207082317598282328618, 6.73933817755199891664261293165, 7.949886587238886812849391663884, 8.417998782571534870749689347403, 9.220805991549512824169296684717, 10.35698240395337473760883098579

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