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Properties

Label 2-29-29.28-c1-0-1
Degree $2$
Conductor $29$
Sign $0.557 + 0.830i$
Analytic cond. $0.231566$
Root an. cond. $0.481213$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.23i·2-s + 2.23i·3-s − 3.00·4-s − 3·5-s + 5.00·6-s + 2·7-s + 2.23i·8-s − 2.00·9-s + 6.70i·10-s − 2.23i·11-s − 6.70i·12-s − 13-s − 4.47i·14-s − 6.70i·15-s − 0.999·16-s + 4.47i·17-s + ⋯
L(s)  = 1  − 1.58i·2-s + 1.29i·3-s − 1.50·4-s − 1.34·5-s + 2.04·6-s + 0.755·7-s + 0.790i·8-s − 0.666·9-s + 2.12i·10-s − 0.674i·11-s − 1.93i·12-s − 0.277·13-s − 1.19i·14-s − 1.73i·15-s − 0.249·16-s + 1.08i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.557 + 0.830i$
Analytic conductor: \(0.231566\)
Root analytic conductor: \(0.481213\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :1/2),\ 0.557 + 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.543733 - 0.289994i\)
\(L(\frac12)\) \(\approx\) \(0.543733 - 0.289994i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (3 + 4.47i)T \)
good2 \( 1 + 2.23iT - 2T^{2} \)
3 \( 1 - 2.23iT - 3T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + 2.23iT - 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 - 4.47iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
31 \( 1 + 6.70iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 4.47iT - 41T^{2} \)
43 \( 1 - 6.70iT - 43T^{2} \)
47 \( 1 + 2.23iT - 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 13.4iT - 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 6.70iT - 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 - 4.47iT - 89T^{2} \)
97 \( 1 + 13.4iT - 97T^{2} \)
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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

−16.99778015087597905058112764402, −15.64798762633737740755367231951, −14.71890083440623415685845527252, −12.86797184750678925981940018018, −11.36178678057320903112498398069, −11.05371410469819666640288849456, −9.643088170761776087576267011655, −8.215421025489560642376724408658, −4.60354096694864069732607307798, −3.55916987895440262001177104646, 4.92263227468655885794947561274, 7.07243239456893174212750743030, 7.45878499055687683322781099288, 8.662965289571010527502610511555, 11.47438908687119495406679323194, 12.63899564306666386775870220650, 14.07619189428567131477170846308, 15.06413860942384192522890506409, 16.06488766482125120963498063711, 17.39588301424158933875677218648

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