Properties

Label 2-29-1.1-c11-0-23
Degree $2$
Conductor $29$
Sign $-1$
Analytic cond. $22.2819$
Root an. cond. $4.72037$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 51.9·2-s + 135.·3-s + 655.·4-s + 3.68e3·5-s + 7.04e3·6-s − 8.37e4·7-s − 7.24e4·8-s − 1.58e5·9-s + 1.91e5·10-s + 1.32e5·11-s + 8.87e4·12-s + 1.93e5·13-s − 4.35e6·14-s + 4.98e5·15-s − 5.10e6·16-s + 2.59e6·17-s − 8.25e6·18-s − 4.52e6·19-s + 2.41e6·20-s − 1.13e7·21-s + 6.91e6·22-s + 7.42e6·23-s − 9.80e6·24-s − 3.52e7·25-s + 1.00e7·26-s − 4.54e7·27-s − 5.48e7·28-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.321·3-s + 0.319·4-s + 0.527·5-s + 0.369·6-s − 1.88·7-s − 0.781·8-s − 0.896·9-s + 0.605·10-s + 0.248·11-s + 0.102·12-s + 0.144·13-s − 2.16·14-s + 0.169·15-s − 1.21·16-s + 0.443·17-s − 1.02·18-s − 0.418·19-s + 0.168·20-s − 0.606·21-s + 0.286·22-s + 0.240·23-s − 0.251·24-s − 0.722·25-s + 0.166·26-s − 0.610·27-s − 0.602·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-1$
Analytic conductor: \(22.2819\)
Root analytic conductor: \(4.72037\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 29,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 - 2.05e7T \)
good2 \( 1 - 51.9T + 2.04e3T^{2} \)
3 \( 1 - 135.T + 1.77e5T^{2} \)
5 \( 1 - 3.68e3T + 4.88e7T^{2} \)
7 \( 1 + 8.37e4T + 1.97e9T^{2} \)
11 \( 1 - 1.32e5T + 2.85e11T^{2} \)
13 \( 1 - 1.93e5T + 1.79e12T^{2} \)
17 \( 1 - 2.59e6T + 3.42e13T^{2} \)
19 \( 1 + 4.52e6T + 1.16e14T^{2} \)
23 \( 1 - 7.42e6T + 9.52e14T^{2} \)
31 \( 1 - 4.28e7T + 2.54e16T^{2} \)
37 \( 1 + 8.25e7T + 1.77e17T^{2} \)
41 \( 1 + 7.25e8T + 5.50e17T^{2} \)
43 \( 1 + 1.14e8T + 9.29e17T^{2} \)
47 \( 1 + 3.13e8T + 2.47e18T^{2} \)
53 \( 1 + 5.04e8T + 9.26e18T^{2} \)
59 \( 1 - 8.68e9T + 3.01e19T^{2} \)
61 \( 1 - 4.85e9T + 4.35e19T^{2} \)
67 \( 1 + 1.44e10T + 1.22e20T^{2} \)
71 \( 1 + 2.67e10T + 2.31e20T^{2} \)
73 \( 1 - 2.46e10T + 3.13e20T^{2} \)
79 \( 1 + 3.86e10T + 7.47e20T^{2} \)
83 \( 1 - 1.74e9T + 1.28e21T^{2} \)
89 \( 1 + 2.10e10T + 2.77e21T^{2} \)
97 \( 1 - 1.02e11T + 7.15e21T^{2} \)
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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

−13.73461466737885331146698169999, −13.05800797386137042065729143241, −11.90692889638362597964207325864, −9.939247948850449774230660193363, −8.867280244989054460593507827430, −6.53749369163539220760769832257, −5.63368550940696544343466728405, −3.69745526060338199260234282921, −2.72658368189996727621729630620, 0, 2.72658368189996727621729630620, 3.69745526060338199260234282921, 5.63368550940696544343466728405, 6.53749369163539220760769832257, 8.867280244989054460593507827430, 9.939247948850449774230660193363, 11.90692889638362597964207325864, 13.05800797386137042065729143241, 13.73461466737885331146698169999

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