Properties

Label 2-1441-1.1-c1-0-54
Degree $2$
Conductor $1441$
Sign $1$
Analytic cond. $11.5064$
Root an. cond. $3.39211$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.67·2-s − 1.26·3-s + 5.13·4-s − 0.362·5-s − 3.37·6-s + 0.0617·7-s + 8.38·8-s − 1.40·9-s − 0.968·10-s + 11-s − 6.48·12-s + 3.38·13-s + 0.165·14-s + 0.457·15-s + 12.1·16-s + 3.44·17-s − 3.75·18-s + 1.54·19-s − 1.86·20-s − 0.0779·21-s + 2.67·22-s + 6.24·23-s − 10.5·24-s − 4.86·25-s + 9.04·26-s + 5.56·27-s + 0.317·28-s + ⋯
L(s)  = 1  + 1.88·2-s − 0.728·3-s + 2.56·4-s − 0.162·5-s − 1.37·6-s + 0.0233·7-s + 2.96·8-s − 0.468·9-s − 0.306·10-s + 0.301·11-s − 1.87·12-s + 0.939·13-s + 0.0441·14-s + 0.118·15-s + 3.03·16-s + 0.834·17-s − 0.885·18-s + 0.353·19-s − 0.416·20-s − 0.0170·21-s + 0.569·22-s + 1.30·23-s − 2.16·24-s − 0.973·25-s + 1.77·26-s + 1.07·27-s + 0.0599·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $1$
Analytic conductor: \(11.5064\)
Root analytic conductor: \(3.39211\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1441,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.478888520\)
\(L(\frac12)\) \(\approx\) \(4.478888520\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - T \)
131 \( 1 + T \)
good2 \( 1 - 2.67T + 2T^{2} \)
3 \( 1 + 1.26T + 3T^{2} \)
5 \( 1 + 0.362T + 5T^{2} \)
7 \( 1 - 0.0617T + 7T^{2} \)
13 \( 1 - 3.38T + 13T^{2} \)
17 \( 1 - 3.44T + 17T^{2} \)
19 \( 1 - 1.54T + 19T^{2} \)
23 \( 1 - 6.24T + 23T^{2} \)
29 \( 1 - 2.65T + 29T^{2} \)
31 \( 1 + 3.16T + 31T^{2} \)
37 \( 1 - 7.14T + 37T^{2} \)
41 \( 1 + 5.84T + 41T^{2} \)
43 \( 1 - 1.61T + 43T^{2} \)
47 \( 1 + 7.80T + 47T^{2} \)
53 \( 1 - 4.33T + 53T^{2} \)
59 \( 1 + 1.45T + 59T^{2} \)
61 \( 1 + 4.89T + 61T^{2} \)
67 \( 1 - 6.96T + 67T^{2} \)
71 \( 1 + 14.2T + 71T^{2} \)
73 \( 1 - 3.77T + 73T^{2} \)
79 \( 1 + 4.66T + 79T^{2} \)
83 \( 1 + 9.72T + 83T^{2} \)
89 \( 1 + 1.46T + 89T^{2} \)
97 \( 1 + 9.53T + 97T^{2} \)
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

−9.841564239066327514655277913275, −8.516230289110070608217028251713, −7.52544756961045933134333046973, −6.63405985991649492464055030741, −5.97953114277694517721890688538, −5.37745217893018735868324000137, −4.58870557920760831179363376842, −3.58560926999516488792442809686, −2.89381385060688321905020563684, −1.37482637578199326880136989974, 1.37482637578199326880136989974, 2.89381385060688321905020563684, 3.58560926999516488792442809686, 4.58870557920760831179363376842, 5.37745217893018735868324000137, 5.97953114277694517721890688538, 6.63405985991649492464055030741, 7.52544756961045933134333046973, 8.516230289110070608217028251713, 9.841564239066327514655277913275

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