Properties

Label 2-1441-1.1-c1-0-35
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.363·2-s − 2.50·3-s − 1.86·4-s − 1.53·5-s + 0.913·6-s − 1.87·7-s + 1.40·8-s + 3.29·9-s + 0.557·10-s + 11-s + 4.68·12-s + 0.782·13-s + 0.680·14-s + 3.84·15-s + 3.22·16-s − 1.92·17-s − 1.20·18-s − 0.355·19-s + 2.86·20-s + 4.69·21-s − 0.363·22-s + 4.07·23-s − 3.53·24-s − 2.65·25-s − 0.284·26-s − 0.748·27-s + 3.49·28-s + ⋯
L(s)  = 1  − 0.257·2-s − 1.44·3-s − 0.933·4-s − 0.685·5-s + 0.372·6-s − 0.707·7-s + 0.497·8-s + 1.09·9-s + 0.176·10-s + 0.301·11-s + 1.35·12-s + 0.217·13-s + 0.181·14-s + 0.993·15-s + 0.805·16-s − 0.466·17-s − 0.282·18-s − 0.0815·19-s + 0.640·20-s + 1.02·21-s − 0.0775·22-s + 0.849·23-s − 0.720·24-s − 0.530·25-s − 0.0558·26-s − 0.144·27-s + 0.660·28-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: \(1\)
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 \( 1 + 0.363T + 2T^{2} \)
3 \( 1 + 2.50T + 3T^{2} \)
5 \( 1 + 1.53T + 5T^{2} \)
7 \( 1 + 1.87T + 7T^{2} \)
13 \( 1 - 0.782T + 13T^{2} \)
17 \( 1 + 1.92T + 17T^{2} \)
19 \( 1 + 0.355T + 19T^{2} \)
23 \( 1 - 4.07T + 23T^{2} \)
29 \( 1 - 3.98T + 29T^{2} \)
31 \( 1 - 8.49T + 31T^{2} \)
37 \( 1 - 9.50T + 37T^{2} \)
41 \( 1 + 1.85T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 - 8.35T + 47T^{2} \)
53 \( 1 - 4.19T + 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 - 8.93T + 67T^{2} \)
71 \( 1 + 15.4T + 71T^{2} \)
73 \( 1 - 0.658T + 73T^{2} \)
79 \( 1 + 17.3T + 79T^{2} \)
83 \( 1 + 3.79T + 83T^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 - 8.38T + 97T^{2} \)
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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

−9.186353678789301985863091531637, −8.397395945863854697950993184144, −7.46900096758442805795872559606, −6.49532797552277502847250388143, −5.92280028325903594481791359642, −4.77946917526818582050200851910, −4.33572313892995473405636430500, −3.15816258501196990370114781207, −1.03896257906143815812630632021, 0, 1.03896257906143815812630632021, 3.15816258501196990370114781207, 4.33572313892995473405636430500, 4.77946917526818582050200851910, 5.92280028325903594481791359642, 6.49532797552277502847250388143, 7.46900096758442805795872559606, 8.397395945863854697950993184144, 9.186353678789301985863091531637

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