Properties

Label 2-29-1.1-c17-0-8
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  + 361.·2-s − 2.25e3·3-s − 683.·4-s − 4.33e5·5-s − 8.15e5·6-s − 1.41e7·7-s − 4.75e7·8-s − 1.24e8·9-s − 1.56e8·10-s − 2.28e8·11-s + 1.54e6·12-s + 4.30e9·13-s − 5.09e9·14-s + 9.79e8·15-s − 1.70e10·16-s + 3.47e10·17-s − 4.47e10·18-s + 4.54e10·19-s + 2.95e8·20-s + 3.19e10·21-s − 8.26e10·22-s + 7.89e10·23-s + 1.07e11·24-s − 5.75e11·25-s + 1.55e12·26-s + 5.72e11·27-s + 9.64e9·28-s + ⋯
L(s)  = 1  + 0.997·2-s − 0.198·3-s − 0.00521·4-s − 0.495·5-s − 0.198·6-s − 0.925·7-s − 1.00·8-s − 0.960·9-s − 0.494·10-s − 0.321·11-s + 0.00103·12-s + 1.46·13-s − 0.923·14-s + 0.0986·15-s − 0.994·16-s + 1.20·17-s − 0.957·18-s + 0.614·19-s + 0.00258·20-s + 0.184·21-s − 0.320·22-s + 0.210·23-s + 0.199·24-s − 0.753·25-s + 1.45·26-s + 0.389·27-s + 0.00482·28-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

\(L(9)\) \(\approx\) \(1.747696094\)
\(L(\frac12)\) \(\approx\) \(1.747696094\)
\(L(\frac{19}{2})\) 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 \( 1 - 361.T + 1.31e5T^{2} \)
3 \( 1 + 2.25e3T + 1.29e8T^{2} \)
5 \( 1 + 4.33e5T + 7.62e11T^{2} \)
7 \( 1 + 1.41e7T + 2.32e14T^{2} \)
11 \( 1 + 2.28e8T + 5.05e17T^{2} \)
13 \( 1 - 4.30e9T + 8.65e18T^{2} \)
17 \( 1 - 3.47e10T + 8.27e20T^{2} \)
19 \( 1 - 4.54e10T + 5.48e21T^{2} \)
23 \( 1 - 7.89e10T + 1.41e23T^{2} \)
31 \( 1 + 1.07e12T + 2.25e25T^{2} \)
37 \( 1 - 2.06e13T + 4.56e26T^{2} \)
41 \( 1 + 5.29e12T + 2.61e27T^{2} \)
43 \( 1 - 4.56e13T + 5.87e27T^{2} \)
47 \( 1 + 5.43e13T + 2.66e28T^{2} \)
53 \( 1 + 1.49e14T + 2.05e29T^{2} \)
59 \( 1 + 2.63e14T + 1.27e30T^{2} \)
61 \( 1 - 6.02e14T + 2.24e30T^{2} \)
67 \( 1 - 1.53e15T + 1.10e31T^{2} \)
71 \( 1 - 9.32e15T + 2.96e31T^{2} \)
73 \( 1 + 9.89e14T + 4.74e31T^{2} \)
79 \( 1 - 6.42e15T + 1.81e32T^{2} \)
83 \( 1 - 2.58e16T + 4.21e32T^{2} \)
89 \( 1 + 4.98e16T + 1.37e33T^{2} \)
97 \( 1 - 4.02e16T + 5.95e33T^{2} \)
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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

−13.33001989982167236079232616330, −12.25875175185861857449648894319, −11.20380114256012857816644997963, −9.502488055490368184569698927170, −8.147896565354933502207436579684, −6.27373107462884513666836145652, −5.41591567100428066750583971355, −3.77305916007213745143950418152, −3.02360808091452198506893553632, −0.62504940108394009249786280625, 0.62504940108394009249786280625, 3.02360808091452198506893553632, 3.77305916007213745143950418152, 5.41591567100428066750583971355, 6.27373107462884513666836145652, 8.147896565354933502207436579684, 9.502488055490368184569698927170, 11.20380114256012857816644997963, 12.25875175185861857449648894319, 13.33001989982167236079232616330

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