Properties

Label 2-29-1.1-c5-0-8
Degree $2$
Conductor $29$
Sign $-1$
Analytic cond. $4.65113$
Root an. cond. $2.15664$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.863·2-s + 7.07·3-s − 31.2·4-s − 44.4·5-s − 6.10·6-s − 36.7·7-s + 54.6·8-s − 192.·9-s + 38.4·10-s − 302.·11-s − 221.·12-s − 373.·13-s + 31.7·14-s − 314.·15-s + 952.·16-s + 280.·17-s + 166.·18-s + 1.37e3·19-s + 1.39e3·20-s − 259.·21-s + 261.·22-s + 1.86e3·23-s + 386.·24-s − 1.14e3·25-s + 322.·26-s − 3.08e3·27-s + 1.14e3·28-s + ⋯
L(s)  = 1  − 0.152·2-s + 0.453·3-s − 0.976·4-s − 0.795·5-s − 0.0692·6-s − 0.283·7-s + 0.301·8-s − 0.794·9-s + 0.121·10-s − 0.753·11-s − 0.443·12-s − 0.612·13-s + 0.0432·14-s − 0.361·15-s + 0.930·16-s + 0.235·17-s + 0.121·18-s + 0.871·19-s + 0.777·20-s − 0.128·21-s + 0.114·22-s + 0.733·23-s + 0.136·24-s − 0.367·25-s + 0.0935·26-s − 0.814·27-s + 0.276·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-1$
Analytic conductor: \(4.65113\)
Root analytic conductor: \(2.15664\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 29,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + 841T \)
good2 \( 1 + 0.863T + 32T^{2} \)
3 \( 1 - 7.07T + 243T^{2} \)
5 \( 1 + 44.4T + 3.12e3T^{2} \)
7 \( 1 + 36.7T + 1.68e4T^{2} \)
11 \( 1 + 302.T + 1.61e5T^{2} \)
13 \( 1 + 373.T + 3.71e5T^{2} \)
17 \( 1 - 280.T + 1.41e6T^{2} \)
19 \( 1 - 1.37e3T + 2.47e6T^{2} \)
23 \( 1 - 1.86e3T + 6.43e6T^{2} \)
31 \( 1 - 1.47e3T + 2.86e7T^{2} \)
37 \( 1 + 1.17e4T + 6.93e7T^{2} \)
41 \( 1 + 2.17e3T + 1.15e8T^{2} \)
43 \( 1 + 9.67e3T + 1.47e8T^{2} \)
47 \( 1 + 1.59e4T + 2.29e8T^{2} \)
53 \( 1 - 2.43e4T + 4.18e8T^{2} \)
59 \( 1 - 3.63e4T + 7.14e8T^{2} \)
61 \( 1 + 2.23e4T + 8.44e8T^{2} \)
67 \( 1 + 5.48e4T + 1.35e9T^{2} \)
71 \( 1 - 2.77e4T + 1.80e9T^{2} \)
73 \( 1 - 3.16e4T + 2.07e9T^{2} \)
79 \( 1 + 5.53e4T + 3.07e9T^{2} \)
83 \( 1 + 4.68e4T + 3.93e9T^{2} \)
89 \( 1 - 2.56e3T + 5.58e9T^{2} \)
97 \( 1 - 3.49e4T + 8.58e9T^{2} \)
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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

−15.35340511458343306645370433307, −14.21983429633934115766648761941, −13.11715264233845930210191032761, −11.72950213981280341246007594859, −10.01668248323960477023189601115, −8.700123517443802261234511953102, −7.62839149562236999932527429138, −5.16879042930089851983957118009, −3.33815165503288978586231114877, 0, 3.33815165503288978586231114877, 5.16879042930089851983957118009, 7.62839149562236999932527429138, 8.700123517443802261234511953102, 10.01668248323960477023189601115, 11.72950213981280341246007594859, 13.11715264233845930210191032761, 14.21983429633934115766648761941, 15.35340511458343306645370433307

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