Properties

Label 2-252-1.1-c11-0-16
Degree $2$
Conductor $252$
Sign $-1$
Analytic cond. $193.622$
Root an. cond. $13.9148$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9.93e3·5-s + 1.68e4·7-s − 7.28e5·11-s + 7.43e5·13-s − 3.15e6·17-s + 1.12e7·19-s − 9.99e5·23-s + 4.98e7·25-s + 1.65e8·29-s − 2.48e8·31-s − 1.66e8·35-s + 4.28e8·37-s + 8.47e8·41-s + 2.27e8·43-s − 1.26e9·47-s + 2.82e8·49-s − 5.00e9·53-s + 7.23e9·55-s + 7.66e9·59-s + 1.03e10·61-s − 7.38e9·65-s − 4.42e9·67-s + 1.58e10·71-s + 2.59e10·73-s − 1.22e10·77-s + 1.54e9·79-s − 3.30e10·83-s + ⋯
L(s)  = 1  − 1.42·5-s + 0.377·7-s − 1.36·11-s + 0.555·13-s − 0.539·17-s + 1.03·19-s − 0.0323·23-s + 1.02·25-s + 1.49·29-s − 1.55·31-s − 0.537·35-s + 1.01·37-s + 1.14·41-s + 0.236·43-s − 0.804·47-s + 0.142·49-s − 1.64·53-s + 1.93·55-s + 1.39·59-s + 1.56·61-s − 0.789·65-s − 0.400·67-s + 1.04·71-s + 1.46·73-s − 0.515·77-s + 0.0563·79-s − 0.921·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(193.622\)
Root analytic conductor: \(13.9148\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 252,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - 1.68e4T \)
good5 \( 1 + 9.93e3T + 4.88e7T^{2} \)
11 \( 1 + 7.28e5T + 2.85e11T^{2} \)
13 \( 1 - 7.43e5T + 1.79e12T^{2} \)
17 \( 1 + 3.15e6T + 3.42e13T^{2} \)
19 \( 1 - 1.12e7T + 1.16e14T^{2} \)
23 \( 1 + 9.99e5T + 9.52e14T^{2} \)
29 \( 1 - 1.65e8T + 1.22e16T^{2} \)
31 \( 1 + 2.48e8T + 2.54e16T^{2} \)
37 \( 1 - 4.28e8T + 1.77e17T^{2} \)
41 \( 1 - 8.47e8T + 5.50e17T^{2} \)
43 \( 1 - 2.27e8T + 9.29e17T^{2} \)
47 \( 1 + 1.26e9T + 2.47e18T^{2} \)
53 \( 1 + 5.00e9T + 9.26e18T^{2} \)
59 \( 1 - 7.66e9T + 3.01e19T^{2} \)
61 \( 1 - 1.03e10T + 4.35e19T^{2} \)
67 \( 1 + 4.42e9T + 1.22e20T^{2} \)
71 \( 1 - 1.58e10T + 2.31e20T^{2} \)
73 \( 1 - 2.59e10T + 3.13e20T^{2} \)
79 \( 1 - 1.54e9T + 7.47e20T^{2} \)
83 \( 1 + 3.30e10T + 1.28e21T^{2} \)
89 \( 1 + 4.41e9T + 2.77e21T^{2} \)
97 \( 1 + 4.12e10T + 7.15e21T^{2} \)
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

−9.626470359504987201357301970055, −8.319443097808592607909403348666, −7.87839131160116467919038725656, −6.92836636642676057420836856410, −5.49376245667337452807366594688, −4.54426810963010913156716670786, −3.55434205700489161658954549588, −2.51257861866304224778746375325, −0.974688311950984533678453465735, 0, 0.974688311950984533678453465735, 2.51257861866304224778746375325, 3.55434205700489161658954549588, 4.54426810963010913156716670786, 5.49376245667337452807366594688, 6.92836636642676057420836856410, 7.87839131160116467919038725656, 8.319443097808592607909403348666, 9.626470359504987201357301970055

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