Properties

Label 2-29-1.1-c11-0-24
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  + 78.6·2-s − 108.·3-s + 4.13e3·4-s − 1.24e4·5-s − 8.56e3·6-s − 230.·7-s + 1.64e5·8-s − 1.65e5·9-s − 9.80e5·10-s − 2.67e4·11-s − 4.50e5·12-s − 1.35e6·13-s − 1.81e4·14-s + 1.35e6·15-s + 4.44e6·16-s + 3.19e6·17-s − 1.29e7·18-s − 9.37e6·19-s − 5.15e7·20-s + 2.50e4·21-s − 2.10e6·22-s − 1.17e7·23-s − 1.78e7·24-s + 1.06e8·25-s − 1.06e8·26-s + 3.73e7·27-s − 9.52e5·28-s + ⋯
L(s)  = 1  + 1.73·2-s − 0.258·3-s + 2.01·4-s − 1.78·5-s − 0.449·6-s − 0.00517·7-s + 1.77·8-s − 0.932·9-s − 3.09·10-s − 0.0500·11-s − 0.522·12-s − 1.01·13-s − 0.00899·14-s + 0.461·15-s + 1.05·16-s + 0.546·17-s − 1.62·18-s − 0.868·19-s − 3.60·20-s + 0.00134·21-s − 0.0869·22-s − 0.379·23-s − 0.458·24-s + 2.18·25-s − 1.75·26-s + 0.500·27-s − 0.0104·28-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 \( 1 - 78.6T + 2.04e3T^{2} \)
3 \( 1 + 108.T + 1.77e5T^{2} \)
5 \( 1 + 1.24e4T + 4.88e7T^{2} \)
7 \( 1 + 230.T + 1.97e9T^{2} \)
11 \( 1 + 2.67e4T + 2.85e11T^{2} \)
13 \( 1 + 1.35e6T + 1.79e12T^{2} \)
17 \( 1 - 3.19e6T + 3.42e13T^{2} \)
19 \( 1 + 9.37e6T + 1.16e14T^{2} \)
23 \( 1 + 1.17e7T + 9.52e14T^{2} \)
31 \( 1 - 1.69e8T + 2.54e16T^{2} \)
37 \( 1 + 6.21e8T + 1.77e17T^{2} \)
41 \( 1 + 9.80e8T + 5.50e17T^{2} \)
43 \( 1 - 1.55e9T + 9.29e17T^{2} \)
47 \( 1 - 7.66e8T + 2.47e18T^{2} \)
53 \( 1 + 1.24e7T + 9.26e18T^{2} \)
59 \( 1 - 3.94e9T + 3.01e19T^{2} \)
61 \( 1 + 6.43e9T + 4.35e19T^{2} \)
67 \( 1 - 4.26e9T + 1.22e20T^{2} \)
71 \( 1 - 2.70e10T + 2.31e20T^{2} \)
73 \( 1 + 8.00e9T + 3.13e20T^{2} \)
79 \( 1 - 4.92e10T + 7.47e20T^{2} \)
83 \( 1 + 7.07e10T + 1.28e21T^{2} \)
89 \( 1 + 8.54e10T + 2.77e21T^{2} \)
97 \( 1 + 4.48e10T + 7.15e21T^{2} \)
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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

−14.10330150199299706971816461747, −12.38606184078883198513088307563, −11.94962510220527705975460246904, −10.87787806218312478591958761284, −8.146289745391791324854354461984, −6.79461943550228118063377265196, −5.17757132729296728175770668576, −4.03667951182717984168562811124, −2.83652279303155170286532931185, 0, 2.83652279303155170286532931185, 4.03667951182717984168562811124, 5.17757132729296728175770668576, 6.79461943550228118063377265196, 8.146289745391791324854354461984, 10.87787806218312478591958761284, 11.94962510220527705975460246904, 12.38606184078883198513088307563, 14.10330150199299706971816461747

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