Properties

Label 2-29-1.1-c11-0-22
Degree $2$
Conductor $29$
Sign $-1$
Analytic cond. $22.2819$
Root an. cond. $4.72037$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 30.9·2-s + 543.·3-s − 1.09e3·4-s − 6.16e3·5-s + 1.68e4·6-s + 8.45e3·7-s − 9.70e4·8-s + 1.18e5·9-s − 1.90e5·10-s − 4.56e5·11-s − 5.94e5·12-s − 1.58e6·13-s + 2.61e5·14-s − 3.35e6·15-s − 7.63e5·16-s − 2.76e6·17-s + 3.67e6·18-s + 1.02e7·19-s + 6.73e6·20-s + 4.59e6·21-s − 1.41e7·22-s − 4.57e6·23-s − 5.28e7·24-s − 1.08e7·25-s − 4.90e7·26-s − 3.17e7·27-s − 9.23e6·28-s + ⋯
L(s)  = 1  + 0.683·2-s + 1.29·3-s − 0.533·4-s − 0.882·5-s + 0.882·6-s + 0.190·7-s − 1.04·8-s + 0.670·9-s − 0.602·10-s − 0.855·11-s − 0.689·12-s − 1.18·13-s + 0.129·14-s − 1.14·15-s − 0.182·16-s − 0.471·17-s + 0.457·18-s + 0.947·19-s + 0.470·20-s + 0.245·21-s − 0.584·22-s − 0.148·23-s − 1.35·24-s − 0.221·25-s − 0.809·26-s − 0.426·27-s − 0.101·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-1$
Analytic conductor: \(22.2819\)
Root analytic conductor: \(4.72037\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 29,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 - 2.05e7T \)
good2 \( 1 - 30.9T + 2.04e3T^{2} \)
3 \( 1 - 543.T + 1.77e5T^{2} \)
5 \( 1 + 6.16e3T + 4.88e7T^{2} \)
7 \( 1 - 8.45e3T + 1.97e9T^{2} \)
11 \( 1 + 4.56e5T + 2.85e11T^{2} \)
13 \( 1 + 1.58e6T + 1.79e12T^{2} \)
17 \( 1 + 2.76e6T + 3.42e13T^{2} \)
19 \( 1 - 1.02e7T + 1.16e14T^{2} \)
23 \( 1 + 4.57e6T + 9.52e14T^{2} \)
31 \( 1 + 1.86e7T + 2.54e16T^{2} \)
37 \( 1 + 3.49e8T + 1.77e17T^{2} \)
41 \( 1 - 4.05e8T + 5.50e17T^{2} \)
43 \( 1 + 5.45e8T + 9.29e17T^{2} \)
47 \( 1 - 2.57e8T + 2.47e18T^{2} \)
53 \( 1 - 2.86e9T + 9.26e18T^{2} \)
59 \( 1 - 9.78e8T + 3.01e19T^{2} \)
61 \( 1 - 6.10e9T + 4.35e19T^{2} \)
67 \( 1 - 2.11e10T + 1.22e20T^{2} \)
71 \( 1 + 5.71e9T + 2.31e20T^{2} \)
73 \( 1 + 3.20e10T + 3.13e20T^{2} \)
79 \( 1 - 2.00e9T + 7.47e20T^{2} \)
83 \( 1 - 1.74e10T + 1.28e21T^{2} \)
89 \( 1 - 2.66e9T + 2.77e21T^{2} \)
97 \( 1 + 4.71e10T + 7.15e21T^{2} \)
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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

−14.04161573780152795440139811747, −13.00718793871031578887570507093, −11.78463838407143598119765796095, −9.777882260381649389371174773647, −8.512998534354754737114248230754, −7.51688957921755997620197025626, −5.10064609151974116820436709395, −3.77826710354917225734068048621, −2.60993016213978565941083056424, 0, 2.60993016213978565941083056424, 3.77826710354917225734068048621, 5.10064609151974116820436709395, 7.51688957921755997620197025626, 8.512998534354754737114248230754, 9.777882260381649389371174773647, 11.78463838407143598119765796095, 13.00718793871031578887570507093, 14.04161573780152795440139811747

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