Properties

Label 2-29-1.1-c7-0-6
Degree $2$
Conductor $29$
Sign $1$
Analytic cond. $9.05916$
Root an. cond. $3.00984$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 22.0·2-s + 73.8·3-s + 359.·4-s + 376.·5-s − 1.63e3·6-s + 647.·7-s − 5.09e3·8-s + 3.27e3·9-s − 8.31e3·10-s − 917.·11-s + 2.65e4·12-s + 5.43e3·13-s − 1.42e4·14-s + 2.78e4·15-s + 6.65e4·16-s − 2.37e4·17-s − 7.22e4·18-s − 2.98e4·19-s + 1.35e5·20-s + 4.78e4·21-s + 2.02e4·22-s + 3.41e4·23-s − 3.76e5·24-s + 6.38e4·25-s − 1.20e5·26-s + 8.03e4·27-s + 2.32e5·28-s + ⋯
L(s)  = 1  − 1.95·2-s + 1.58·3-s + 2.80·4-s + 1.34·5-s − 3.08·6-s + 0.713·7-s − 3.52·8-s + 1.49·9-s − 2.62·10-s − 0.207·11-s + 4.43·12-s + 0.686·13-s − 1.39·14-s + 2.13·15-s + 4.06·16-s − 1.17·17-s − 2.92·18-s − 0.999·19-s + 3.78·20-s + 1.12·21-s + 0.405·22-s + 0.585·23-s − 5.56·24-s + 0.817·25-s − 1.33·26-s + 0.785·27-s + 2.00·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $1$
Analytic conductor: \(9.05916\)
Root analytic conductor: \(3.00984\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(1.591541538\)
\(L(\frac12)\) \(\approx\) \(1.591541538\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + 2.43e4T \)
good2 \( 1 + 22.0T + 128T^{2} \)
3 \( 1 - 73.8T + 2.18e3T^{2} \)
5 \( 1 - 376.T + 7.81e4T^{2} \)
7 \( 1 - 647.T + 8.23e5T^{2} \)
11 \( 1 + 917.T + 1.94e7T^{2} \)
13 \( 1 - 5.43e3T + 6.27e7T^{2} \)
17 \( 1 + 2.37e4T + 4.10e8T^{2} \)
19 \( 1 + 2.98e4T + 8.93e8T^{2} \)
23 \( 1 - 3.41e4T + 3.40e9T^{2} \)
31 \( 1 - 2.15e5T + 2.75e10T^{2} \)
37 \( 1 - 6.11e4T + 9.49e10T^{2} \)
41 \( 1 + 2.02e5T + 1.94e11T^{2} \)
43 \( 1 - 4.36e5T + 2.71e11T^{2} \)
47 \( 1 + 1.14e6T + 5.06e11T^{2} \)
53 \( 1 - 2.37e5T + 1.17e12T^{2} \)
59 \( 1 + 1.56e6T + 2.48e12T^{2} \)
61 \( 1 - 2.82e6T + 3.14e12T^{2} \)
67 \( 1 + 1.28e6T + 6.06e12T^{2} \)
71 \( 1 - 3.48e6T + 9.09e12T^{2} \)
73 \( 1 - 6.81e4T + 1.10e13T^{2} \)
79 \( 1 + 7.70e5T + 1.92e13T^{2} \)
83 \( 1 + 3.66e6T + 2.71e13T^{2} \)
89 \( 1 + 3.22e5T + 4.42e13T^{2} \)
97 \( 1 + 2.17e6T + 8.07e13T^{2} \)
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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

−15.65734631116173729923546449560, −14.63709100431646572742975819464, −13.25585039433183607121477608111, −11.00317921631840998837667445658, −9.840064628745964611540206417727, −8.873578035316858422390202804959, −8.152086056524574424019852006278, −6.55289928463314350364201972441, −2.52903148605884788536018562128, −1.60666565104229070117190091702, 1.60666565104229070117190091702, 2.52903148605884788536018562128, 6.55289928463314350364201972441, 8.152086056524574424019852006278, 8.873578035316858422390202804959, 9.840064628745964611540206417727, 11.00317921631840998837667445658, 13.25585039433183607121477608111, 14.63709100431646572742975819464, 15.65734631116173729923546449560

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