Properties

Label 2-29-1.1-c3-0-5
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  − 2.41·2-s − 0.757·3-s − 2.17·4-s − 10.6·5-s + 1.82·6-s − 22.1·7-s + 24.5·8-s − 26.4·9-s + 25.7·10-s + 39.3·11-s + 1.64·12-s + 23.7·13-s + 53.4·14-s + 8.07·15-s − 41.9·16-s + 4.54·17-s + 63.7·18-s − 155.·19-s + 23.1·20-s + 16.7·21-s − 94.9·22-s − 41.8·23-s − 18.5·24-s − 11.4·25-s − 57.3·26-s + 40.4·27-s + 48.0·28-s + ⋯
L(s)  = 1  − 0.853·2-s − 0.145·3-s − 0.271·4-s − 0.953·5-s + 0.124·6-s − 1.19·7-s + 1.08·8-s − 0.978·9-s + 0.813·10-s + 1.07·11-s + 0.0395·12-s + 0.507·13-s + 1.02·14-s + 0.138·15-s − 0.654·16-s + 0.0648·17-s + 0.835·18-s − 1.87·19-s + 0.258·20-s + 0.174·21-s − 0.920·22-s − 0.379·23-s − 0.158·24-s − 0.0914·25-s − 0.432·26-s + 0.288·27-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 \( 1 + 2.41T + 8T^{2} \)
3 \( 1 + 0.757T + 27T^{2} \)
5 \( 1 + 10.6T + 125T^{2} \)
7 \( 1 + 22.1T + 343T^{2} \)
11 \( 1 - 39.3T + 1.33e3T^{2} \)
13 \( 1 - 23.7T + 2.19e3T^{2} \)
17 \( 1 - 4.54T + 4.91e3T^{2} \)
19 \( 1 + 155.T + 6.85e3T^{2} \)
23 \( 1 + 41.8T + 1.21e4T^{2} \)
31 \( 1 + 57.9T + 2.97e4T^{2} \)
37 \( 1 - 235.T + 5.06e4T^{2} \)
41 \( 1 + 175.T + 6.89e4T^{2} \)
43 \( 1 + 402.T + 7.95e4T^{2} \)
47 \( 1 - 227.T + 1.03e5T^{2} \)
53 \( 1 - 673.T + 1.48e5T^{2} \)
59 \( 1 + 800.T + 2.05e5T^{2} \)
61 \( 1 + 222.T + 2.26e5T^{2} \)
67 \( 1 + 524.T + 3.00e5T^{2} \)
71 \( 1 + 281.T + 3.57e5T^{2} \)
73 \( 1 - 1.22e3T + 3.89e5T^{2} \)
79 \( 1 - 611.T + 4.93e5T^{2} \)
83 \( 1 - 515.T + 5.71e5T^{2} \)
89 \( 1 + 358.T + 7.04e5T^{2} \)
97 \( 1 - 829.T + 9.12e5T^{2} \)
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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

−16.46976932620168704908211552687, −14.99362885371431321175005214260, −13.49720619638542510200395714815, −12.06699832388332366568260854880, −10.72109520463557624017405549539, −9.233686056546093846184361494433, −8.235515118369237279416647585400, −6.45363169599506477439245978795, −3.92457225982003136766281733809, 0, 3.92457225982003136766281733809, 6.45363169599506477439245978795, 8.235515118369237279416647585400, 9.233686056546093846184361494433, 10.72109520463557624017405549539, 12.06699832388332366568260854880, 13.49720619638542510200395714815, 14.99362885371431321175005214260, 16.46976932620168704908211552687

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