Properties

Label 2-1441-1.1-c1-0-14
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  − 2.46·2-s − 2.74·3-s + 4.08·4-s + 0.488·5-s + 6.78·6-s − 1.50·7-s − 5.15·8-s + 4.55·9-s − 1.20·10-s + 11-s − 11.2·12-s − 1.24·13-s + 3.70·14-s − 1.34·15-s + 4.54·16-s + 3.29·17-s − 11.2·18-s + 4.48·19-s + 1.99·20-s + 4.12·21-s − 2.46·22-s + 2.86·23-s + 14.1·24-s − 4.76·25-s + 3.07·26-s − 4.27·27-s − 6.14·28-s + ⋯
L(s)  = 1  − 1.74·2-s − 1.58·3-s + 2.04·4-s + 0.218·5-s + 2.76·6-s − 0.567·7-s − 1.82·8-s + 1.51·9-s − 0.381·10-s + 0.301·11-s − 3.24·12-s − 0.345·13-s + 0.990·14-s − 0.346·15-s + 1.13·16-s + 0.799·17-s − 2.64·18-s + 1.02·19-s + 0.447·20-s + 0.900·21-s − 0.526·22-s + 0.597·23-s + 2.89·24-s − 0.952·25-s + 0.602·26-s − 0.822·27-s − 1.16·28-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

\(L(1)\) \(\approx\) \(0.3263814465\)
\(L(\frac12)\) \(\approx\) \(0.3263814465\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.46T + 2T^{2} \)
3 \( 1 + 2.74T + 3T^{2} \)
5 \( 1 - 0.488T + 5T^{2} \)
7 \( 1 + 1.50T + 7T^{2} \)
13 \( 1 + 1.24T + 13T^{2} \)
17 \( 1 - 3.29T + 17T^{2} \)
19 \( 1 - 4.48T + 19T^{2} \)
23 \( 1 - 2.86T + 23T^{2} \)
29 \( 1 + 6.33T + 29T^{2} \)
31 \( 1 - 7.17T + 31T^{2} \)
37 \( 1 + 6.61T + 37T^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 - 2.12T + 43T^{2} \)
47 \( 1 + 8.05T + 47T^{2} \)
53 \( 1 + 2.36T + 53T^{2} \)
59 \( 1 + 1.23T + 59T^{2} \)
61 \( 1 + 12.8T + 61T^{2} \)
67 \( 1 + 7.36T + 67T^{2} \)
71 \( 1 + 1.51T + 71T^{2} \)
73 \( 1 + 10.5T + 73T^{2} \)
79 \( 1 - 9.64T + 79T^{2} \)
83 \( 1 - 2.52T + 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + 9.73T + 97T^{2} \)
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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

−9.679030788307486553796400083043, −9.036590875815633888418403457928, −7.74262649154241800328088575703, −7.28017873968845388880068905655, −6.31634172039404873373576290571, −5.84554891189254301252211489965, −4.75840340225696837605493742719, −3.16648265592704387863038212194, −1.63202910269640410666683401133, −0.59322275014551408390959607183, 0.59322275014551408390959607183, 1.63202910269640410666683401133, 3.16648265592704387863038212194, 4.75840340225696837605493742719, 5.84554891189254301252211489965, 6.31634172039404873373576290571, 7.28017873968845388880068905655, 7.74262649154241800328088575703, 9.036590875815633888418403457928, 9.679030788307486553796400083043

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