Properties

Label 2-29-1.1-c1-0-0
Degree $2$
Conductor $29$
Sign $1$
Analytic cond. $0.231566$
Root an. cond. $0.481213$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.41·2-s + 2.41·3-s + 3.82·4-s − 5-s − 5.82·6-s − 2.82·7-s − 4.41·8-s + 2.82·9-s + 2.41·10-s − 0.414·11-s + 9.24·12-s − 3.82·13-s + 6.82·14-s − 2.41·15-s + 2.99·16-s + 0.828·17-s − 6.82·18-s + 6·19-s − 3.82·20-s − 6.82·21-s + 0.999·22-s + 3.65·23-s − 10.6·24-s − 4·25-s + 9.24·26-s − 0.414·27-s − 10.8·28-s + ⋯
L(s)  = 1  − 1.70·2-s + 1.39·3-s + 1.91·4-s − 0.447·5-s − 2.37·6-s − 1.06·7-s − 1.56·8-s + 0.942·9-s + 0.763·10-s − 0.124·11-s + 2.66·12-s − 1.06·13-s + 1.82·14-s − 0.623·15-s + 0.749·16-s + 0.200·17-s − 1.60·18-s + 1.37·19-s − 0.856·20-s − 1.49·21-s + 0.213·22-s + 0.762·23-s − 2.17·24-s − 0.800·25-s + 1.81·26-s − 0.0797·27-s − 2.04·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $1$
Analytic conductor: \(0.231566\)
Root analytic conductor: \(0.481213\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4520178616\)
\(L(\frac12)\) \(\approx\) \(0.4520178616\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 - T \)
good2 \( 1 + 2.41T + 2T^{2} \)
3 \( 1 - 2.41T + 3T^{2} \)
5 \( 1 + T + 5T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 + 0.414T + 11T^{2} \)
13 \( 1 + 3.82T + 13T^{2} \)
17 \( 1 - 0.828T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 - 3.65T + 23T^{2} \)
31 \( 1 - 10.0T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 4.48T + 41T^{2} \)
43 \( 1 - 3.58T + 43T^{2} \)
47 \( 1 + 3.24T + 47T^{2} \)
53 \( 1 - 9.48T + 53T^{2} \)
59 \( 1 + 3.65T + 59T^{2} \)
61 \( 1 + 4.82T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 + 8.82T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + 2.41T + 79T^{2} \)
83 \( 1 - 7.65T + 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 - 4.48T + 97T^{2} \)
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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

−17.31343986241515149330655261806, −16.05888388043654018981880321944, −15.23224358305790267581443761926, −13.66732976667500941952922868953, −11.93782397100020205133590566736, −10.04252502469782990793395660236, −9.347008880241771359683944351195, −8.111694777874299421477288589426, −7.11407190598914430481785081554, −2.92477068592233051723034627885, 2.92477068592233051723034627885, 7.11407190598914430481785081554, 8.111694777874299421477288589426, 9.347008880241771359683944351195, 10.04252502469782990793395660236, 11.93782397100020205133590566736, 13.66732976667500941952922868953, 15.23224358305790267581443761926, 16.05888388043654018981880321944, 17.31343986241515149330655261806

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