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Properties

Label 2-29-29.28-c9-0-17
Degree $2$
Conductor $29$
Sign $-0.430 + 0.902i$
Analytic cond. $14.9360$
Root an. cond. $3.86471$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 25.3i·2-s − 226. i·3-s − 129.·4-s − 952.·5-s + 5.72e3·6-s + 6.45e3·7-s + 9.69e3i·8-s − 3.14e4·9-s − 2.41e4i·10-s − 7.21e4i·11-s + 2.91e4i·12-s − 1.90e5·13-s + 1.63e5i·14-s + 2.15e5i·15-s − 3.11e5·16-s − 5.86e4i·17-s + ⋯
L(s)  = 1  + 1.11i·2-s − 1.61i·3-s − 0.252·4-s − 0.681·5-s + 1.80·6-s + 1.01·7-s + 0.836i·8-s − 1.59·9-s − 0.762i·10-s − 1.48i·11-s + 0.406i·12-s − 1.84·13-s + 1.13i·14-s + 1.09i·15-s − 1.18·16-s − 0.170i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.430 + 0.902i$
Analytic conductor: \(14.9360\)
Root analytic conductor: \(3.86471\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :9/2),\ -0.430 + 0.902i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.511579 - 0.810551i\)
\(L(\frac12)\) \(\approx\) \(0.511579 - 0.810551i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-1.63e6 + 3.43e6i)T \)
good2 \( 1 - 25.3iT - 512T^{2} \)
3 \( 1 + 226. iT - 1.96e4T^{2} \)
5 \( 1 + 952.T + 1.95e6T^{2} \)
7 \( 1 - 6.45e3T + 4.03e7T^{2} \)
11 \( 1 + 7.21e4iT - 2.35e9T^{2} \)
13 \( 1 + 1.90e5T + 1.06e10T^{2} \)
17 \( 1 + 5.86e4iT - 1.18e11T^{2} \)
19 \( 1 + 5.24e5iT - 3.22e11T^{2} \)
23 \( 1 + 1.18e6T + 1.80e12T^{2} \)
31 \( 1 - 8.71e6iT - 2.64e13T^{2} \)
37 \( 1 + 5.56e6iT - 1.29e14T^{2} \)
41 \( 1 + 4.38e6iT - 3.27e14T^{2} \)
43 \( 1 + 3.96e6iT - 5.02e14T^{2} \)
47 \( 1 + 1.62e7iT - 1.11e15T^{2} \)
53 \( 1 - 2.97e6T + 3.29e15T^{2} \)
59 \( 1 - 1.66e8T + 8.66e15T^{2} \)
61 \( 1 + 1.63e8iT - 1.16e16T^{2} \)
67 \( 1 + 1.87e7T + 2.72e16T^{2} \)
71 \( 1 - 2.59e8T + 4.58e16T^{2} \)
73 \( 1 - 3.06e8iT - 5.88e16T^{2} \)
79 \( 1 - 3.09e7iT - 1.19e17T^{2} \)
83 \( 1 - 7.54e8T + 1.86e17T^{2} \)
89 \( 1 + 3.38e8iT - 3.50e17T^{2} \)
97 \( 1 - 3.80e8iT - 7.60e17T^{2} \)
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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

−14.50453167393913207807457149532, −13.79401267757471837904414813756, −12.10078686610677663997741621248, −11.34249965179657019855853335885, −8.383562321828424401380417047108, −7.71971121668883724553802353378, −6.73804055999644231393496785457, −5.25409932056241747731674069179, −2.29968133259746257404244158265, −0.35985426302598464804741125404, 2.19389233338052498154469441072, 3.98067933935640133463481407117, 4.80380764231782029870829160718, 7.68106515667148183315521959573, 9.664429533251780891117527805123, 10.24922624735211710527125100306, 11.55199562496908888704861906011, 12.28708449320132640644145760469, 14.67061457153706956233345559892, 15.21211363661405203047023071777

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