Properties

Label 2-1441-1.1-c1-0-104
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.28·2-s + 0.625·3-s + 3.22·4-s − 3.41·5-s + 1.43·6-s − 3.04·7-s + 2.79·8-s − 2.60·9-s − 7.80·10-s + 11-s + 2.01·12-s − 0.0108·13-s − 6.95·14-s − 2.13·15-s − 0.0590·16-s + 4.75·17-s − 5.96·18-s − 6.00·19-s − 11.0·20-s − 1.90·21-s + 2.28·22-s − 6.25·23-s + 1.74·24-s + 6.65·25-s − 0.0247·26-s − 3.51·27-s − 9.80·28-s + ⋯
L(s)  = 1  + 1.61·2-s + 0.361·3-s + 1.61·4-s − 1.52·5-s + 0.584·6-s − 1.15·7-s + 0.988·8-s − 0.869·9-s − 2.46·10-s + 0.301·11-s + 0.582·12-s − 0.00300·13-s − 1.85·14-s − 0.551·15-s − 0.0147·16-s + 1.15·17-s − 1.40·18-s − 1.37·19-s − 2.46·20-s − 0.415·21-s + 0.487·22-s − 1.30·23-s + 0.357·24-s + 1.33·25-s − 0.00485·26-s − 0.675·27-s − 1.85·28-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: \(1\)
Selberg data: \((2,\ 1441,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.28T + 2T^{2} \)
3 \( 1 - 0.625T + 3T^{2} \)
5 \( 1 + 3.41T + 5T^{2} \)
7 \( 1 + 3.04T + 7T^{2} \)
13 \( 1 + 0.0108T + 13T^{2} \)
17 \( 1 - 4.75T + 17T^{2} \)
19 \( 1 + 6.00T + 19T^{2} \)
23 \( 1 + 6.25T + 23T^{2} \)
29 \( 1 - 6.23T + 29T^{2} \)
31 \( 1 + 4.73T + 31T^{2} \)
37 \( 1 + 7.75T + 37T^{2} \)
41 \( 1 - 7.20T + 41T^{2} \)
43 \( 1 - 2.12T + 43T^{2} \)
47 \( 1 + 5.17T + 47T^{2} \)
53 \( 1 - 0.876T + 53T^{2} \)
59 \( 1 + 2.17T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 + 0.594T + 67T^{2} \)
71 \( 1 + 9.32T + 71T^{2} \)
73 \( 1 - 6.92T + 73T^{2} \)
79 \( 1 + 8.92T + 79T^{2} \)
83 \( 1 - 5.23T + 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 - 5.22T + 97T^{2} \)
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

−8.935642441801921464095706583791, −8.191645034833578210092698863000, −7.32696327673519036896702583438, −6.42506999119704936483660933175, −5.81338287747293283851290894881, −4.67240995641887845987794665778, −3.69621751567879016709372118293, −3.48818335254007163228164724642, −2.45231956548808721105571933351, 0, 2.45231956548808721105571933351, 3.48818335254007163228164724642, 3.69621751567879016709372118293, 4.67240995641887845987794665778, 5.81338287747293283851290894881, 6.42506999119704936483660933175, 7.32696327673519036896702583438, 8.191645034833578210092698863000, 8.935642441801921464095706583791

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