Properties

Label 2-29-1.1-c17-0-6
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  − 687.·2-s − 2.13e4·3-s + 3.41e5·4-s − 1.16e6·5-s + 1.46e7·6-s − 9.37e6·7-s − 1.44e8·8-s + 3.27e8·9-s + 8.04e8·10-s + 8.41e8·11-s − 7.29e9·12-s + 4.06e9·13-s + 6.44e9·14-s + 2.50e10·15-s + 5.46e10·16-s + 1.44e10·17-s − 2.25e11·18-s − 2.26e9·19-s − 3.99e11·20-s + 2.00e11·21-s − 5.78e11·22-s − 2.37e11·23-s + 3.08e12·24-s + 6.05e11·25-s − 2.79e12·26-s − 4.24e12·27-s − 3.20e12·28-s + ⋯
L(s)  = 1  − 1.89·2-s − 1.88·3-s + 2.60·4-s − 1.33·5-s + 3.57·6-s − 0.614·7-s − 3.04·8-s + 2.53·9-s + 2.54·10-s + 1.18·11-s − 4.89·12-s + 1.38·13-s + 1.16·14-s + 2.51·15-s + 3.17·16-s + 0.501·17-s − 4.81·18-s − 0.0305·19-s − 3.48·20-s + 1.15·21-s − 2.24·22-s − 0.631·23-s + 5.72·24-s + 0.794·25-s − 2.62·26-s − 2.89·27-s − 1.60·28-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

\(L(9)\) \(\approx\) \(0.2976010878\)
\(L(\frac12)\) \(\approx\) \(0.2976010878\)
\(L(\frac{19}{2})\) 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 \( 1 + 687.T + 1.31e5T^{2} \)
3 \( 1 + 2.13e4T + 1.29e8T^{2} \)
5 \( 1 + 1.16e6T + 7.62e11T^{2} \)
7 \( 1 + 9.37e6T + 2.32e14T^{2} \)
11 \( 1 - 8.41e8T + 5.05e17T^{2} \)
13 \( 1 - 4.06e9T + 8.65e18T^{2} \)
17 \( 1 - 1.44e10T + 8.27e20T^{2} \)
19 \( 1 + 2.26e9T + 5.48e21T^{2} \)
23 \( 1 + 2.37e11T + 1.41e23T^{2} \)
31 \( 1 - 7.30e12T + 2.25e25T^{2} \)
37 \( 1 - 1.73e12T + 4.56e26T^{2} \)
41 \( 1 + 2.43e13T + 2.61e27T^{2} \)
43 \( 1 - 8.01e13T + 5.87e27T^{2} \)
47 \( 1 - 9.47e13T + 2.66e28T^{2} \)
53 \( 1 + 5.06e14T + 2.05e29T^{2} \)
59 \( 1 - 1.89e15T + 1.27e30T^{2} \)
61 \( 1 - 5.95e14T + 2.24e30T^{2} \)
67 \( 1 - 7.78e14T + 1.10e31T^{2} \)
71 \( 1 + 2.17e15T + 2.96e31T^{2} \)
73 \( 1 - 1.29e16T + 4.74e31T^{2} \)
79 \( 1 - 1.99e16T + 1.81e32T^{2} \)
83 \( 1 + 2.62e16T + 4.21e32T^{2} \)
89 \( 1 + 3.29e15T + 1.37e33T^{2} \)
97 \( 1 + 5.04e16T + 5.95e33T^{2} \)
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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

−12.25940039003946467305472462458, −11.60872521580043853098744702152, −10.86541641468212196409386751818, −9.716498672599012343388179043325, −8.170132208507554277732135411260, −6.85102773087983214522161094889, −6.15151670093812082149306281121, −3.83853606804548913391173091367, −1.20183801054753012196095268175, −0.53900941982851237518251216268, 0.53900941982851237518251216268, 1.20183801054753012196095268175, 3.83853606804548913391173091367, 6.15151670093812082149306281121, 6.85102773087983214522161094889, 8.170132208507554277732135411260, 9.716498672599012343388179043325, 10.86541641468212196409386751818, 11.60872521580043853098744702152, 12.25940039003946467305472462458

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