Properties

Label 2-29-1.1-c5-0-1
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.60·2-s − 13.2·3-s − 25.2·4-s + 69.0·5-s + 34.6·6-s + 156.·7-s + 149.·8-s − 66.5·9-s − 179.·10-s + 424.·11-s + 334.·12-s + 44.7·13-s − 407.·14-s − 916.·15-s + 418.·16-s + 1.38e3·17-s + 173.·18-s − 1.11e3·19-s − 1.73e3·20-s − 2.07e3·21-s − 1.10e3·22-s − 39.2·23-s − 1.98e3·24-s + 1.63e3·25-s − 116.·26-s + 4.11e3·27-s − 3.94e3·28-s + ⋯
L(s)  = 1  − 0.460·2-s − 0.852·3-s − 0.787·4-s + 1.23·5-s + 0.392·6-s + 1.20·7-s + 0.823·8-s − 0.273·9-s − 0.568·10-s + 1.05·11-s + 0.671·12-s + 0.0734·13-s − 0.556·14-s − 1.05·15-s + 0.408·16-s + 1.15·17-s + 0.126·18-s − 0.711·19-s − 0.972·20-s − 1.02·21-s − 0.487·22-s − 0.0154·23-s − 0.701·24-s + 0.523·25-s − 0.0338·26-s + 1.08·27-s − 0.951·28-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: \(0\)
Selberg data: \((2,\ 29,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.052426940\)
\(L(\frac12)\) \(\approx\) \(1.052426940\)
\(L(\frac{7}{2})\) 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 \( 1 + 2.60T + 32T^{2} \)
3 \( 1 + 13.2T + 243T^{2} \)
5 \( 1 - 69.0T + 3.12e3T^{2} \)
7 \( 1 - 156.T + 1.68e4T^{2} \)
11 \( 1 - 424.T + 1.61e5T^{2} \)
13 \( 1 - 44.7T + 3.71e5T^{2} \)
17 \( 1 - 1.38e3T + 1.41e6T^{2} \)
19 \( 1 + 1.11e3T + 2.47e6T^{2} \)
23 \( 1 + 39.2T + 6.43e6T^{2} \)
31 \( 1 - 8.68e3T + 2.86e7T^{2} \)
37 \( 1 - 2.45e3T + 6.93e7T^{2} \)
41 \( 1 + 1.24e4T + 1.15e8T^{2} \)
43 \( 1 - 2.13e4T + 1.47e8T^{2} \)
47 \( 1 + 7.82e3T + 2.29e8T^{2} \)
53 \( 1 + 3.39e4T + 4.18e8T^{2} \)
59 \( 1 + 2.70e4T + 7.14e8T^{2} \)
61 \( 1 - 7.57e3T + 8.44e8T^{2} \)
67 \( 1 + 5.43e4T + 1.35e9T^{2} \)
71 \( 1 - 8.08e4T + 1.80e9T^{2} \)
73 \( 1 + 4.55e4T + 2.07e9T^{2} \)
79 \( 1 - 6.80e4T + 3.07e9T^{2} \)
83 \( 1 - 3.24e4T + 3.93e9T^{2} \)
89 \( 1 + 3.12e4T + 5.58e9T^{2} \)
97 \( 1 + 1.08e5T + 8.58e9T^{2} \)
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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

−16.84215054602189070822384374162, −14.50165188295766752930682818630, −13.83614253489762613154213746525, −12.16370318813549759247735687346, −10.80204985572839245858462906308, −9.590002721741401440668123849708, −8.269496714659495483899171804385, −6.07865110183826768303300378339, −4.80895890003283281513462284786, −1.27543251167081629088689745430, 1.27543251167081629088689745430, 4.80895890003283281513462284786, 6.07865110183826768303300378339, 8.269496714659495483899171804385, 9.590002721741401440668123849708, 10.80204985572839245858462906308, 12.16370318813549759247735687346, 13.83614253489762613154213746525, 14.50165188295766752930682818630, 16.84215054602189070822384374162

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