Properties

Label 2-29-1.1-c11-0-14
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  − 85.7·2-s + 234.·3-s + 5.30e3·4-s − 1.50e3·5-s − 2.00e4·6-s − 7.06e3·7-s − 2.78e5·8-s − 1.22e5·9-s + 1.29e5·10-s + 5.98e5·11-s + 1.24e6·12-s − 2.84e5·13-s + 6.06e5·14-s − 3.52e5·15-s + 1.30e7·16-s + 6.02e6·17-s + 1.04e7·18-s − 1.04e7·19-s − 7.98e6·20-s − 1.65e6·21-s − 5.12e7·22-s + 3.07e7·23-s − 6.53e7·24-s − 4.65e7·25-s + 2.43e7·26-s − 7.01e7·27-s − 3.74e7·28-s + ⋯
L(s)  = 1  − 1.89·2-s + 0.556·3-s + 2.58·4-s − 0.215·5-s − 1.05·6-s − 0.158·7-s − 3.00·8-s − 0.690·9-s + 0.408·10-s + 1.12·11-s + 1.44·12-s − 0.212·13-s + 0.301·14-s − 0.120·15-s + 3.10·16-s + 1.02·17-s + 1.30·18-s − 0.964·19-s − 0.557·20-s − 0.0885·21-s − 2.12·22-s + 0.995·23-s − 1.67·24-s − 0.953·25-s + 0.402·26-s − 0.940·27-s − 0.411·28-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 \( 1 + 85.7T + 2.04e3T^{2} \)
3 \( 1 - 234.T + 1.77e5T^{2} \)
5 \( 1 + 1.50e3T + 4.88e7T^{2} \)
7 \( 1 + 7.06e3T + 1.97e9T^{2} \)
11 \( 1 - 5.98e5T + 2.85e11T^{2} \)
13 \( 1 + 2.84e5T + 1.79e12T^{2} \)
17 \( 1 - 6.02e6T + 3.42e13T^{2} \)
19 \( 1 + 1.04e7T + 1.16e14T^{2} \)
23 \( 1 - 3.07e7T + 9.52e14T^{2} \)
31 \( 1 - 2.00e7T + 2.54e16T^{2} \)
37 \( 1 + 3.29e8T + 1.77e17T^{2} \)
41 \( 1 + 1.33e8T + 5.50e17T^{2} \)
43 \( 1 + 1.51e8T + 9.29e17T^{2} \)
47 \( 1 - 5.65e8T + 2.47e18T^{2} \)
53 \( 1 + 5.83e9T + 9.26e18T^{2} \)
59 \( 1 + 8.59e9T + 3.01e19T^{2} \)
61 \( 1 + 1.25e10T + 4.35e19T^{2} \)
67 \( 1 - 1.31e10T + 1.22e20T^{2} \)
71 \( 1 + 4.99e9T + 2.31e20T^{2} \)
73 \( 1 - 3.78e9T + 3.13e20T^{2} \)
79 \( 1 + 3.61e10T + 7.47e20T^{2} \)
83 \( 1 + 3.39e10T + 1.28e21T^{2} \)
89 \( 1 - 8.21e10T + 2.77e21T^{2} \)
97 \( 1 + 4.18e10T + 7.15e21T^{2} \)
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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

−14.38927506067912947247367683235, −12.14378141254838401887767827033, −11.04499088988373359604136556513, −9.657427912189066388969121608348, −8.760759543085982668322423480501, −7.71397899481067044100030098968, −6.32595914978853565714014771931, −3.13775074915110377815035613746, −1.57562838929040543115709607227, 0, 1.57562838929040543115709607227, 3.13775074915110377815035613746, 6.32595914978853565714014771931, 7.71397899481067044100030098968, 8.760759543085982668322423480501, 9.657427912189066388969121608348, 11.04499088988373359604136556513, 12.14378141254838401887767827033, 14.38927506067912947247367683235

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