Properties

Label 2-29-1.1-c3-0-6
Degree $2$
Conductor $29$
Sign $-1$
Analytic cond. $1.71105$
Root an. cond. $1.30807$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.414·2-s − 9.24·3-s − 7.82·4-s + 0.656·5-s − 3.82·6-s + 6.14·7-s − 6.55·8-s + 58.4·9-s + 0.272·10-s − 65.3·11-s + 72.3·12-s − 49.7·13-s + 2.54·14-s − 6.07·15-s + 59.9·16-s + 55.4·17-s + 24.2·18-s − 64.7·19-s − 5.14·20-s − 56.7·21-s − 27.0·22-s + 93.8·23-s + 60.5·24-s − 124.·25-s − 20.6·26-s − 290.·27-s − 48.0·28-s + ⋯
L(s)  = 1  + 0.146·2-s − 1.77·3-s − 0.978·4-s + 0.0587·5-s − 0.260·6-s + 0.331·7-s − 0.289·8-s + 2.16·9-s + 0.00860·10-s − 1.79·11-s + 1.74·12-s − 1.06·13-s + 0.0485·14-s − 0.104·15-s + 0.936·16-s + 0.791·17-s + 0.316·18-s − 0.781·19-s − 0.0574·20-s − 0.589·21-s − 0.262·22-s + 0.851·23-s + 0.515·24-s − 0.996·25-s − 0.155·26-s − 2.07·27-s − 0.324·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-1$
Analytic conductor: \(1.71105\)
Root analytic conductor: \(1.30807\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 29,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 - 29T \)
good2 \( 1 - 0.414T + 8T^{2} \)
3 \( 1 + 9.24T + 27T^{2} \)
5 \( 1 - 0.656T + 125T^{2} \)
7 \( 1 - 6.14T + 343T^{2} \)
11 \( 1 + 65.3T + 1.33e3T^{2} \)
13 \( 1 + 49.7T + 2.19e3T^{2} \)
17 \( 1 - 55.4T + 4.91e3T^{2} \)
19 \( 1 + 64.7T + 6.85e3T^{2} \)
23 \( 1 - 93.8T + 1.21e4T^{2} \)
31 \( 1 + 236.T + 2.97e4T^{2} \)
37 \( 1 - 76.8T + 5.06e4T^{2} \)
41 \( 1 - 215.T + 6.89e4T^{2} \)
43 \( 1 - 80.8T + 7.95e4T^{2} \)
47 \( 1 + 357.T + 1.03e5T^{2} \)
53 \( 1 - 328.T + 1.48e5T^{2} \)
59 \( 1 + 99.2T + 2.05e5T^{2} \)
61 \( 1 + 725.T + 2.26e5T^{2} \)
67 \( 1 - 844.T + 3.00e5T^{2} \)
71 \( 1 + 378.T + 3.57e5T^{2} \)
73 \( 1 + 581.T + 3.89e5T^{2} \)
79 \( 1 + 353.T + 4.93e5T^{2} \)
83 \( 1 - 696.T + 5.71e5T^{2} \)
89 \( 1 - 1.11e3T + 7.04e5T^{2} \)
97 \( 1 + 805.T + 9.12e5T^{2} \)
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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

−16.33833750590260599028789669791, −14.90157465963589865550929850725, −13.12577067531635209909531596123, −12.36325042905113069824494992125, −10.89075172801174481768107751876, −9.862476516189287789103102992686, −7.66321302669574866883891850884, −5.62777663248966123170236433844, −4.79240736397582967434846963710, 0, 4.79240736397582967434846963710, 5.62777663248966123170236433844, 7.66321302669574866883891850884, 9.862476516189287789103102992686, 10.89075172801174481768107751876, 12.36325042905113069824494992125, 13.12577067531635209909531596123, 14.90157465963589865550929850725, 16.33833750590260599028789669791

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