Properties

Label 2-29-1.1-c17-0-3
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  − 3.42·2-s − 6.75e3·3-s − 1.31e5·4-s + 5.11e5·5-s + 2.31e4·6-s − 1.86e7·7-s + 8.98e5·8-s − 8.35e7·9-s − 1.75e6·10-s − 3.73e8·11-s + 8.85e8·12-s − 1.25e9·13-s + 6.38e7·14-s − 3.45e9·15-s + 1.71e10·16-s − 2.47e10·17-s + 2.86e8·18-s − 1.19e11·19-s − 6.70e10·20-s + 1.25e11·21-s + 1.28e9·22-s − 5.78e11·23-s − 6.07e9·24-s − 5.00e11·25-s + 4.29e9·26-s + 1.43e12·27-s + 2.44e12·28-s + ⋯
L(s)  = 1  − 0.00947·2-s − 0.594·3-s − 0.999·4-s + 0.586·5-s + 0.00563·6-s − 1.22·7-s + 0.0189·8-s − 0.646·9-s − 0.00555·10-s − 0.524·11-s + 0.594·12-s − 0.426·13-s + 0.0115·14-s − 0.348·15-s + 0.999·16-s − 0.861·17-s + 0.00612·18-s − 1.61·19-s − 0.585·20-s + 0.726·21-s + 0.00497·22-s − 1.53·23-s − 0.0112·24-s − 0.656·25-s + 0.00403·26-s + 0.978·27-s + 1.22·28-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

\(L(9)\) \(\approx\) \(0.1491808300\)
\(L(\frac12)\) \(\approx\) \(0.1491808300\)
\(L(\frac{19}{2})\) 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 \( 1 + 3.42T + 1.31e5T^{2} \)
3 \( 1 + 6.75e3T + 1.29e8T^{2} \)
5 \( 1 - 5.11e5T + 7.62e11T^{2} \)
7 \( 1 + 1.86e7T + 2.32e14T^{2} \)
11 \( 1 + 3.73e8T + 5.05e17T^{2} \)
13 \( 1 + 1.25e9T + 8.65e18T^{2} \)
17 \( 1 + 2.47e10T + 8.27e20T^{2} \)
19 \( 1 + 1.19e11T + 5.48e21T^{2} \)
23 \( 1 + 5.78e11T + 1.41e23T^{2} \)
31 \( 1 + 5.22e11T + 2.25e25T^{2} \)
37 \( 1 + 1.62e13T + 4.56e26T^{2} \)
41 \( 1 + 1.89e13T + 2.61e27T^{2} \)
43 \( 1 - 8.14e13T + 5.87e27T^{2} \)
47 \( 1 - 1.46e14T + 2.66e28T^{2} \)
53 \( 1 + 2.00e14T + 2.05e29T^{2} \)
59 \( 1 - 9.31e14T + 1.27e30T^{2} \)
61 \( 1 - 3.72e14T + 2.24e30T^{2} \)
67 \( 1 - 3.15e15T + 1.10e31T^{2} \)
71 \( 1 - 9.53e15T + 2.96e31T^{2} \)
73 \( 1 + 3.79e14T + 4.74e31T^{2} \)
79 \( 1 + 1.81e16T + 1.81e32T^{2} \)
83 \( 1 + 2.82e16T + 4.21e32T^{2} \)
89 \( 1 - 5.52e16T + 1.37e33T^{2} \)
97 \( 1 + 5.77e16T + 5.95e33T^{2} \)
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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

−13.22284348435888325630532410633, −12.32562983395225021255820185665, −10.57006071900947832113889682890, −9.604987670925753875170067199515, −8.414612289190671515347043402548, −6.43141040865645416461054984588, −5.50208692410022934049925261979, −4.04379410011877862097501676103, −2.38426747087963273025736109858, −0.20727280477487331390619760773, 0.20727280477487331390619760773, 2.38426747087963273025736109858, 4.04379410011877862097501676103, 5.50208692410022934049925261979, 6.43141040865645416461054984588, 8.414612289190671515347043402548, 9.604987670925753875170067199515, 10.57006071900947832113889682890, 12.32562983395225021255820185665, 13.22284348435888325630532410633

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