Properties

Label 2-29-1.1-c11-0-10
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  − 23.7·2-s − 804.·3-s − 1.48e3·4-s + 6.64e3·5-s + 1.90e4·6-s − 6.11e4·7-s + 8.37e4·8-s + 4.69e5·9-s − 1.57e5·10-s + 9.71e4·11-s + 1.19e6·12-s + 2.25e6·13-s + 1.44e6·14-s − 5.34e6·15-s + 1.05e6·16-s − 5.47e6·17-s − 1.11e7·18-s − 1.36e7·19-s − 9.87e6·20-s + 4.91e7·21-s − 2.30e6·22-s + 3.78e7·23-s − 6.73e7·24-s − 4.68e6·25-s − 5.33e7·26-s − 2.34e8·27-s + 9.08e7·28-s + ⋯
L(s)  = 1  − 0.523·2-s − 1.91·3-s − 0.725·4-s + 0.950·5-s + 1.00·6-s − 1.37·7-s + 0.903·8-s + 2.64·9-s − 0.497·10-s + 0.181·11-s + 1.38·12-s + 1.68·13-s + 0.720·14-s − 1.81·15-s + 0.252·16-s − 0.934·17-s − 1.38·18-s − 1.26·19-s − 0.689·20-s + 2.62·21-s − 0.0952·22-s + 1.22·23-s − 1.72·24-s − 0.0960·25-s − 0.881·26-s − 3.15·27-s + 0.997·28-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 \( 1 + 23.7T + 2.04e3T^{2} \)
3 \( 1 + 804.T + 1.77e5T^{2} \)
5 \( 1 - 6.64e3T + 4.88e7T^{2} \)
7 \( 1 + 6.11e4T + 1.97e9T^{2} \)
11 \( 1 - 9.71e4T + 2.85e11T^{2} \)
13 \( 1 - 2.25e6T + 1.79e12T^{2} \)
17 \( 1 + 5.47e6T + 3.42e13T^{2} \)
19 \( 1 + 1.36e7T + 1.16e14T^{2} \)
23 \( 1 - 3.78e7T + 9.52e14T^{2} \)
31 \( 1 - 7.66e7T + 2.54e16T^{2} \)
37 \( 1 - 5.87e8T + 1.77e17T^{2} \)
41 \( 1 + 5.25e8T + 5.50e17T^{2} \)
43 \( 1 - 1.40e9T + 9.29e17T^{2} \)
47 \( 1 + 1.05e9T + 2.47e18T^{2} \)
53 \( 1 - 1.82e8T + 9.26e18T^{2} \)
59 \( 1 + 4.81e9T + 3.01e19T^{2} \)
61 \( 1 - 1.42e9T + 4.35e19T^{2} \)
67 \( 1 - 1.58e9T + 1.22e20T^{2} \)
71 \( 1 + 7.93e9T + 2.31e20T^{2} \)
73 \( 1 + 2.06e10T + 3.13e20T^{2} \)
79 \( 1 + 3.33e10T + 7.47e20T^{2} \)
83 \( 1 + 4.80e9T + 1.28e21T^{2} \)
89 \( 1 + 3.03e10T + 2.77e21T^{2} \)
97 \( 1 + 2.92e10T + 7.15e21T^{2} \)
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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

−13.27347608957404140265225730778, −12.93991566576337194996697473447, −11.06269473961239626614915259243, −10.17825488612201384184299478612, −9.092927424751655871718432927699, −6.59178505642270038182250248166, −5.89980505746473732002045292730, −4.29778887234999796928351970262, −1.18598569186602746355060952457, 0, 1.18598569186602746355060952457, 4.29778887234999796928351970262, 5.89980505746473732002045292730, 6.59178505642270038182250248166, 9.092927424751655871718432927699, 10.17825488612201384184299478612, 11.06269473961239626614915259243, 12.93991566576337194996697473447, 13.27347608957404140265225730778

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