Properties

Label 2-29-1.1-c11-0-16
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  − 21.9·2-s + 310.·3-s − 1.56e3·4-s − 886.·5-s − 6.82e3·6-s + 4.42e4·7-s + 7.94e4·8-s − 8.08e4·9-s + 1.94e4·10-s − 4.80e5·11-s − 4.85e5·12-s + 7.29e5·13-s − 9.73e5·14-s − 2.75e5·15-s + 1.45e6·16-s + 6.80e6·17-s + 1.77e6·18-s − 1.56e7·19-s + 1.38e6·20-s + 1.37e7·21-s + 1.05e7·22-s − 3.18e7·23-s + 2.46e7·24-s − 4.80e7·25-s − 1.60e7·26-s − 8.00e7·27-s − 6.92e7·28-s + ⋯
L(s)  = 1  − 0.486·2-s + 0.737·3-s − 0.763·4-s − 0.126·5-s − 0.358·6-s + 0.995·7-s + 0.857·8-s − 0.456·9-s + 0.0616·10-s − 0.899·11-s − 0.563·12-s + 0.544·13-s − 0.483·14-s − 0.0935·15-s + 0.346·16-s + 1.16·17-s + 0.221·18-s − 1.44·19-s + 0.0968·20-s + 0.734·21-s + 0.437·22-s − 1.03·23-s + 0.632·24-s − 0.983·25-s − 0.264·26-s − 1.07·27-s − 0.760·28-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 \( 1 + 21.9T + 2.04e3T^{2} \)
3 \( 1 - 310.T + 1.77e5T^{2} \)
5 \( 1 + 886.T + 4.88e7T^{2} \)
7 \( 1 - 4.42e4T + 1.97e9T^{2} \)
11 \( 1 + 4.80e5T + 2.85e11T^{2} \)
13 \( 1 - 7.29e5T + 1.79e12T^{2} \)
17 \( 1 - 6.80e6T + 3.42e13T^{2} \)
19 \( 1 + 1.56e7T + 1.16e14T^{2} \)
23 \( 1 + 3.18e7T + 9.52e14T^{2} \)
31 \( 1 + 3.08e8T + 2.54e16T^{2} \)
37 \( 1 - 4.28e8T + 1.77e17T^{2} \)
41 \( 1 + 1.00e9T + 5.50e17T^{2} \)
43 \( 1 - 3.53e8T + 9.29e17T^{2} \)
47 \( 1 + 2.87e9T + 2.47e18T^{2} \)
53 \( 1 + 1.08e9T + 9.26e18T^{2} \)
59 \( 1 - 9.69e9T + 3.01e19T^{2} \)
61 \( 1 - 2.73e9T + 4.35e19T^{2} \)
67 \( 1 - 5.53e9T + 1.22e20T^{2} \)
71 \( 1 - 1.51e10T + 2.31e20T^{2} \)
73 \( 1 - 2.05e10T + 3.13e20T^{2} \)
79 \( 1 + 2.27e10T + 7.47e20T^{2} \)
83 \( 1 + 6.37e9T + 1.28e21T^{2} \)
89 \( 1 + 2.92e10T + 2.77e21T^{2} \)
97 \( 1 + 1.24e11T + 7.15e21T^{2} \)
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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

−14.09702107604542950663083764859, −12.94425402563309952350110118709, −11.17640987620082984203459106503, −9.841664804433024072473183567546, −8.355183942466229255976878854326, −7.972995242018431303775494057506, −5.40959466469036278463281763038, −3.81557952704054681826734558961, −1.87420858397928547291201097065, 0, 1.87420858397928547291201097065, 3.81557952704054681826734558961, 5.40959466469036278463281763038, 7.972995242018431303775494057506, 8.355183942466229255976878854326, 9.841664804433024072473183567546, 11.17640987620082984203459106503, 12.94425402563309952350110118709, 14.09702107604542950663083764859

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