Properties

Label 2-1441-1.1-c1-0-18
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.68·2-s − 1.38·3-s + 0.836·4-s − 3.96·5-s − 2.33·6-s + 0.750·7-s − 1.95·8-s − 1.06·9-s − 6.68·10-s − 11-s − 1.16·12-s − 0.0294·13-s + 1.26·14-s + 5.51·15-s − 4.97·16-s + 7.46·17-s − 1.80·18-s + 2.54·19-s − 3.31·20-s − 1.04·21-s − 1.68·22-s + 8.13·23-s + 2.72·24-s + 10.7·25-s − 0.0496·26-s + 5.65·27-s + 0.627·28-s + ⋯
L(s)  = 1  + 1.19·2-s − 0.802·3-s + 0.418·4-s − 1.77·5-s − 0.955·6-s + 0.283·7-s − 0.692·8-s − 0.356·9-s − 2.11·10-s − 0.301·11-s − 0.335·12-s − 0.00817·13-s + 0.337·14-s + 1.42·15-s − 1.24·16-s + 1.81·17-s − 0.424·18-s + 0.583·19-s − 0.742·20-s − 0.227·21-s − 0.359·22-s + 1.69·23-s + 0.555·24-s + 2.15·25-s − 0.00973·26-s + 1.08·27-s + 0.118·28-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1441,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.254451074\)
\(L(\frac12)\) \(\approx\) \(1.254451074\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.68T + 2T^{2} \)
3 \( 1 + 1.38T + 3T^{2} \)
5 \( 1 + 3.96T + 5T^{2} \)
7 \( 1 - 0.750T + 7T^{2} \)
13 \( 1 + 0.0294T + 13T^{2} \)
17 \( 1 - 7.46T + 17T^{2} \)
19 \( 1 - 2.54T + 19T^{2} \)
23 \( 1 - 8.13T + 23T^{2} \)
29 \( 1 + 7.66T + 29T^{2} \)
31 \( 1 + 6.50T + 31T^{2} \)
37 \( 1 - 2.24T + 37T^{2} \)
41 \( 1 - 11.9T + 41T^{2} \)
43 \( 1 + 3.54T + 43T^{2} \)
47 \( 1 - 4.31T + 47T^{2} \)
53 \( 1 - 5.31T + 53T^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 - 1.65T + 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 4.85T + 73T^{2} \)
79 \( 1 + 6.42T + 79T^{2} \)
83 \( 1 + 8.46T + 83T^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 - 8.93T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.499772619893033703185242048475, −8.583228506539410988430269869133, −7.64484137210499344682362502635, −7.12676926282264131576911254750, −5.81695261673776188282637355040, −5.29174018252017894525640624565, −4.55703255538506742742999484647, −3.56882716203446142252998843576, −3.04150122734897967087850296491, −0.68986456556299640786247407375, 0.68986456556299640786247407375, 3.04150122734897967087850296491, 3.56882716203446142252998843576, 4.55703255538506742742999484647, 5.29174018252017894525640624565, 5.81695261673776188282637355040, 7.12676926282264131576911254750, 7.64484137210499344682362502635, 8.583228506539410988430269869133, 9.499772619893033703185242048475

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