Properties

Label 2-29-1.1-c9-0-13
Degree $2$
Conductor $29$
Sign $-1$
Analytic cond. $14.9360$
Root an. cond. $3.86471$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 35.6·2-s + 190.·3-s + 759.·4-s − 886.·5-s − 6.79e3·6-s − 3.87e3·7-s − 8.81e3·8-s + 1.66e4·9-s + 3.15e4·10-s + 2.60e4·11-s + 1.44e5·12-s − 6.57e3·13-s + 1.38e5·14-s − 1.68e5·15-s − 7.43e4·16-s − 1.49e5·17-s − 5.92e5·18-s − 1.43e5·19-s − 6.72e5·20-s − 7.38e5·21-s − 9.29e5·22-s − 3.28e5·23-s − 1.67e6·24-s − 1.16e6·25-s + 2.34e5·26-s − 5.84e5·27-s − 2.94e6·28-s + ⋯
L(s)  = 1  − 1.57·2-s + 1.35·3-s + 1.48·4-s − 0.634·5-s − 2.13·6-s − 0.610·7-s − 0.761·8-s + 0.844·9-s + 0.999·10-s + 0.536·11-s + 2.01·12-s − 0.0638·13-s + 0.962·14-s − 0.861·15-s − 0.283·16-s − 0.433·17-s − 1.33·18-s − 0.252·19-s − 0.940·20-s − 0.829·21-s − 0.845·22-s − 0.244·23-s − 1.03·24-s − 0.597·25-s + 0.100·26-s − 0.211·27-s − 0.905·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-1$
Analytic conductor: \(14.9360\)
Root analytic conductor: \(3.86471\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 29,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + 7.07e5T \)
good2 \( 1 + 35.6T + 512T^{2} \)
3 \( 1 - 190.T + 1.96e4T^{2} \)
5 \( 1 + 886.T + 1.95e6T^{2} \)
7 \( 1 + 3.87e3T + 4.03e7T^{2} \)
11 \( 1 - 2.60e4T + 2.35e9T^{2} \)
13 \( 1 + 6.57e3T + 1.06e10T^{2} \)
17 \( 1 + 1.49e5T + 1.18e11T^{2} \)
19 \( 1 + 1.43e5T + 3.22e11T^{2} \)
23 \( 1 + 3.28e5T + 1.80e12T^{2} \)
31 \( 1 + 8.83e6T + 2.64e13T^{2} \)
37 \( 1 - 3.11e6T + 1.29e14T^{2} \)
41 \( 1 + 2.46e7T + 3.27e14T^{2} \)
43 \( 1 + 4.11e7T + 5.02e14T^{2} \)
47 \( 1 - 1.85e7T + 1.11e15T^{2} \)
53 \( 1 + 1.68e7T + 3.29e15T^{2} \)
59 \( 1 - 1.55e8T + 8.66e15T^{2} \)
61 \( 1 - 1.48e8T + 1.16e16T^{2} \)
67 \( 1 + 1.13e8T + 2.72e16T^{2} \)
71 \( 1 - 1.60e8T + 4.58e16T^{2} \)
73 \( 1 + 1.93e8T + 5.88e16T^{2} \)
79 \( 1 - 2.95e8T + 1.19e17T^{2} \)
83 \( 1 - 3.45e8T + 1.86e17T^{2} \)
89 \( 1 + 9.93e8T + 3.50e17T^{2} \)
97 \( 1 - 1.17e9T + 7.60e17T^{2} \)
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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

−14.75461550327117772772490883053, −13.27679443066149296310812981234, −11.51174718435144581211680314761, −9.957436229195675358423885668429, −8.992383058377539193453741041621, −8.112407328406468554420558009966, −6.94932059545327105533932723246, −3.62479041217308518237313867451, −1.95835826654431768371396625325, 0, 1.95835826654431768371396625325, 3.62479041217308518237313867451, 6.94932059545327105533932723246, 8.112407328406468554420558009966, 8.992383058377539193453741041621, 9.957436229195675358423885668429, 11.51174718435144581211680314761, 13.27679443066149296310812981234, 14.75461550327117772772490883053

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