Properties

Label 2-1441-1.1-c1-0-44
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  + 0.665·2-s − 3.11·3-s − 1.55·4-s − 0.365·5-s − 2.07·6-s − 1.41·7-s − 2.36·8-s + 6.71·9-s − 0.243·10-s + 11-s + 4.85·12-s + 2.19·13-s − 0.944·14-s + 1.14·15-s + 1.53·16-s + 7.23·17-s + 4.46·18-s − 2.16·19-s + 0.569·20-s + 4.42·21-s + 0.665·22-s + 2.10·23-s + 7.37·24-s − 4.86·25-s + 1.45·26-s − 11.5·27-s + 2.21·28-s + ⋯
L(s)  = 1  + 0.470·2-s − 1.79·3-s − 0.778·4-s − 0.163·5-s − 0.846·6-s − 0.536·7-s − 0.836·8-s + 2.23·9-s − 0.0770·10-s + 0.301·11-s + 1.40·12-s + 0.608·13-s − 0.252·14-s + 0.294·15-s + 0.384·16-s + 1.75·17-s + 1.05·18-s − 0.497·19-s + 0.127·20-s + 0.965·21-s + 0.141·22-s + 0.439·23-s + 1.50·24-s − 0.973·25-s + 0.286·26-s − 2.22·27-s + 0.417·28-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 \( 1 - 0.665T + 2T^{2} \)
3 \( 1 + 3.11T + 3T^{2} \)
5 \( 1 + 0.365T + 5T^{2} \)
7 \( 1 + 1.41T + 7T^{2} \)
13 \( 1 - 2.19T + 13T^{2} \)
17 \( 1 - 7.23T + 17T^{2} \)
19 \( 1 + 2.16T + 19T^{2} \)
23 \( 1 - 2.10T + 23T^{2} \)
29 \( 1 - 1.53T + 29T^{2} \)
31 \( 1 + 6.46T + 31T^{2} \)
37 \( 1 + 5.05T + 37T^{2} \)
41 \( 1 - 3.54T + 41T^{2} \)
43 \( 1 - 3.24T + 43T^{2} \)
47 \( 1 + 9.24T + 47T^{2} \)
53 \( 1 - 4.01T + 53T^{2} \)
59 \( 1 + 1.83T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 + 7.96T + 67T^{2} \)
71 \( 1 - 0.988T + 71T^{2} \)
73 \( 1 - 0.0955T + 73T^{2} \)
79 \( 1 + 9.70T + 79T^{2} \)
83 \( 1 - 9.43T + 83T^{2} \)
89 \( 1 + 5.27T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−9.428764663168508062158511279441, −8.285661065595731268443367069454, −7.23679686885326622425015053238, −6.31335881467075602082940136521, −5.70638045539072137797043939417, −5.14139003771224399749336576835, −4.12341162096713079988218152029, −3.43286034770348742221721308366, −1.23550367969478907518531935475, 0, 1.23550367969478907518531935475, 3.43286034770348742221721308366, 4.12341162096713079988218152029, 5.14139003771224399749336576835, 5.70638045539072137797043939417, 6.31335881467075602082940136521, 7.23679686885326622425015053238, 8.285661065595731268443367069454, 9.428764663168508062158511279441

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