Properties

Label 2-29-1.1-c17-0-22
Degree $2$
Conductor $29$
Sign $1$
Analytic cond. $53.1344$
Root an. cond. $7.28933$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 645.·2-s + 1.48e4·3-s + 2.85e5·4-s + 9.54e5·5-s − 9.60e6·6-s + 1.80e7·7-s − 9.98e7·8-s + 9.23e7·9-s − 6.16e8·10-s − 1.63e8·11-s + 4.25e9·12-s + 4.33e9·13-s − 1.16e10·14-s + 1.42e10·15-s + 2.70e10·16-s + 5.11e10·17-s − 5.96e10·18-s + 8.96e10·19-s + 2.72e11·20-s + 2.68e11·21-s + 1.05e11·22-s − 2.78e11·23-s − 1.48e12·24-s + 1.48e11·25-s − 2.79e12·26-s − 5.47e11·27-s + 5.16e12·28-s + ⋯
L(s)  = 1  − 1.78·2-s + 1.30·3-s + 2.18·4-s + 1.09·5-s − 2.33·6-s + 1.18·7-s − 2.10·8-s + 0.715·9-s − 1.94·10-s − 0.230·11-s + 2.85·12-s + 1.47·13-s − 2.11·14-s + 1.43·15-s + 1.57·16-s + 1.77·17-s − 1.27·18-s + 1.21·19-s + 2.38·20-s + 1.55·21-s + 0.410·22-s − 0.741·23-s − 2.75·24-s + 0.194·25-s − 2.62·26-s − 0.372·27-s + 2.58·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $1$
Analytic conductor: \(53.1344\)
Root analytic conductor: \(7.28933\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :17/2),\ 1)\)

Particular Values

\(L(9)\) \(\approx\) \(2.422879795\)
\(L(\frac12)\) \(\approx\) \(2.422879795\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 - 5.00e11T \)
good2 \( 1 + 645.T + 1.31e5T^{2} \)
3 \( 1 - 1.48e4T + 1.29e8T^{2} \)
5 \( 1 - 9.54e5T + 7.62e11T^{2} \)
7 \( 1 - 1.80e7T + 2.32e14T^{2} \)
11 \( 1 + 1.63e8T + 5.05e17T^{2} \)
13 \( 1 - 4.33e9T + 8.65e18T^{2} \)
17 \( 1 - 5.11e10T + 8.27e20T^{2} \)
19 \( 1 - 8.96e10T + 5.48e21T^{2} \)
23 \( 1 + 2.78e11T + 1.41e23T^{2} \)
31 \( 1 + 2.36e12T + 2.25e25T^{2} \)
37 \( 1 - 3.03e13T + 4.56e26T^{2} \)
41 \( 1 + 8.45e13T + 2.61e27T^{2} \)
43 \( 1 + 1.94e13T + 5.87e27T^{2} \)
47 \( 1 - 9.65e13T + 2.66e28T^{2} \)
53 \( 1 + 5.94e14T + 2.05e29T^{2} \)
59 \( 1 + 1.80e14T + 1.27e30T^{2} \)
61 \( 1 + 1.33e15T + 2.24e30T^{2} \)
67 \( 1 - 2.44e15T + 1.10e31T^{2} \)
71 \( 1 + 5.66e15T + 2.96e31T^{2} \)
73 \( 1 + 1.53e15T + 4.74e31T^{2} \)
79 \( 1 + 1.72e16T + 1.81e32T^{2} \)
83 \( 1 - 8.38e15T + 4.21e32T^{2} \)
89 \( 1 + 3.85e16T + 1.37e33T^{2} \)
97 \( 1 - 8.15e16T + 5.95e33T^{2} \)
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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

−13.66154327728286488379105205677, −11.53675949485983626950852358164, −10.20929066184906819637543924207, −9.356265295083974552093104643113, −8.291141126189540708165974194902, −7.68407248899596201155442694184, −5.80929489150549642681031008877, −3.14009435515935794966674977015, −1.79514369702171619039792275922, −1.23475542751916058660095677241, 1.23475542751916058660095677241, 1.79514369702171619039792275922, 3.14009435515935794966674977015, 5.80929489150549642681031008877, 7.68407248899596201155442694184, 8.291141126189540708165974194902, 9.356265295083974552093104643113, 10.20929066184906819637543924207, 11.53675949485983626950852358164, 13.66154327728286488379105205677

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