Properties

Label 2-29-1.1-c17-0-14
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  − 237.·2-s − 1.41e4·3-s − 7.45e4·4-s + 1.22e6·5-s + 3.37e6·6-s + 1.05e7·7-s + 4.88e7·8-s + 7.20e7·9-s − 2.90e8·10-s + 9.82e8·11-s + 1.05e9·12-s + 5.20e9·13-s − 2.51e9·14-s − 1.73e10·15-s − 1.86e9·16-s − 2.48e10·17-s − 1.71e10·18-s + 7.35e10·19-s − 9.10e10·20-s − 1.49e11·21-s − 2.33e11·22-s − 4.63e11·23-s − 6.93e11·24-s + 7.29e11·25-s − 1.23e12·26-s + 8.09e11·27-s − 7.86e11·28-s + ⋯
L(s)  = 1  − 0.656·2-s − 1.24·3-s − 0.568·4-s + 1.39·5-s + 0.820·6-s + 0.692·7-s + 1.03·8-s + 0.558·9-s − 0.918·10-s + 1.38·11-s + 0.709·12-s + 1.77·13-s − 0.454·14-s − 1.74·15-s − 0.108·16-s − 0.863·17-s − 0.366·18-s + 0.993·19-s − 0.794·20-s − 0.863·21-s − 0.907·22-s − 1.23·23-s − 1.28·24-s + 0.956·25-s − 1.16·26-s + 0.551·27-s − 0.393·28-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

\(L(9)\) \(\approx\) \(1.405775071\)
\(L(\frac12)\) \(\approx\) \(1.405775071\)
\(L(\frac{19}{2})\) 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 \( 1 + 237.T + 1.31e5T^{2} \)
3 \( 1 + 1.41e4T + 1.29e8T^{2} \)
5 \( 1 - 1.22e6T + 7.62e11T^{2} \)
7 \( 1 - 1.05e7T + 2.32e14T^{2} \)
11 \( 1 - 9.82e8T + 5.05e17T^{2} \)
13 \( 1 - 5.20e9T + 8.65e18T^{2} \)
17 \( 1 + 2.48e10T + 8.27e20T^{2} \)
19 \( 1 - 7.35e10T + 5.48e21T^{2} \)
23 \( 1 + 4.63e11T + 1.41e23T^{2} \)
31 \( 1 - 4.00e12T + 2.25e25T^{2} \)
37 \( 1 - 9.74e12T + 4.56e26T^{2} \)
41 \( 1 - 5.79e13T + 2.61e27T^{2} \)
43 \( 1 + 8.74e13T + 5.87e27T^{2} \)
47 \( 1 - 8.94e13T + 2.66e28T^{2} \)
53 \( 1 - 6.03e14T + 2.05e29T^{2} \)
59 \( 1 - 9.08e14T + 1.27e30T^{2} \)
61 \( 1 + 2.23e15T + 2.24e30T^{2} \)
67 \( 1 - 1.04e15T + 1.10e31T^{2} \)
71 \( 1 - 2.03e15T + 2.96e31T^{2} \)
73 \( 1 + 3.65e15T + 4.74e31T^{2} \)
79 \( 1 - 6.95e15T + 1.81e32T^{2} \)
83 \( 1 + 1.05e16T + 4.21e32T^{2} \)
89 \( 1 + 3.63e16T + 1.37e33T^{2} \)
97 \( 1 - 1.37e17T + 5.95e33T^{2} \)
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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

−13.45509737135551415277588915454, −11.72456431746384482802319028891, −10.72339259313876738591249202581, −9.568205081757098329171617332079, −8.506739263630567185202379251386, −6.46327816078076640464976113180, −5.61238385387607476718305731074, −4.25758292376878231104200351477, −1.58748124921774700233115227453, −0.887972723781716850296972329589, 0.887972723781716850296972329589, 1.58748124921774700233115227453, 4.25758292376878231104200351477, 5.61238385387607476718305731074, 6.46327816078076640464976113180, 8.506739263630567185202379251386, 9.568205081757098329171617332079, 10.72339259313876738591249202581, 11.72456431746384482802319028891, 13.45509737135551415277588915454

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