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Properties

Label 2-29-29.28-c9-0-4
Degree $2$
Conductor $29$
Sign $-0.887 + 0.460i$
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  + 20.3i·2-s + 80.4i·3-s + 98.0·4-s + 535.·5-s − 1.63e3·6-s − 1.06e4·7-s + 1.24e4i·8-s + 1.32e4·9-s + 1.08e4i·10-s + 4.41e4i·11-s + 7.88e3i·12-s − 1.58e5·13-s − 2.16e5i·14-s + 4.30e4i·15-s − 2.02e5·16-s + 1.40e5i·17-s + ⋯
L(s)  = 1  + 0.899i·2-s + 0.573i·3-s + 0.191·4-s + 0.382·5-s − 0.515·6-s − 1.67·7-s + 1.07i·8-s + 0.671·9-s + 0.344i·10-s + 0.908i·11-s + 0.109i·12-s − 1.54·13-s − 1.50i·14-s + 0.219i·15-s − 0.771·16-s + 0.407i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.887 + 0.460i$
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: \((2,\ 29,\ (\ :9/2),\ -0.887 + 0.460i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.273483 - 1.12193i\)
\(L(\frac12)\) \(\approx\) \(0.273483 - 1.12193i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-3.38e6 + 1.75e6i)T \)
good2 \( 1 - 20.3iT - 512T^{2} \)
3 \( 1 - 80.4iT - 1.96e4T^{2} \)
5 \( 1 - 535.T + 1.95e6T^{2} \)
7 \( 1 + 1.06e4T + 4.03e7T^{2} \)
11 \( 1 - 4.41e4iT - 2.35e9T^{2} \)
13 \( 1 + 1.58e5T + 1.06e10T^{2} \)
17 \( 1 - 1.40e5iT - 1.18e11T^{2} \)
19 \( 1 + 8.77e5iT - 3.22e11T^{2} \)
23 \( 1 + 3.30e5T + 1.80e12T^{2} \)
31 \( 1 + 1.48e6iT - 2.64e13T^{2} \)
37 \( 1 - 1.79e7iT - 1.29e14T^{2} \)
41 \( 1 - 2.53e7iT - 3.27e14T^{2} \)
43 \( 1 + 3.46e7iT - 5.02e14T^{2} \)
47 \( 1 - 5.33e7iT - 1.11e15T^{2} \)
53 \( 1 - 5.06e7T + 3.29e15T^{2} \)
59 \( 1 + 1.38e8T + 8.66e15T^{2} \)
61 \( 1 - 1.55e8iT - 1.16e16T^{2} \)
67 \( 1 - 5.38e7T + 2.72e16T^{2} \)
71 \( 1 - 4.97e7T + 4.58e16T^{2} \)
73 \( 1 - 8.08e7iT - 5.88e16T^{2} \)
79 \( 1 + 1.32e8iT - 1.19e17T^{2} \)
83 \( 1 - 5.96e8T + 1.86e17T^{2} \)
89 \( 1 - 2.06e8iT - 3.50e17T^{2} \)
97 \( 1 - 1.61e9iT - 7.60e17T^{2} \)
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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

−15.61018044758636288231969153961, −15.10878730350851065972155226550, −13.42052096577064663091775893491, −12.17378607688810454848032665181, −10.18195468775664480539189087672, −9.451692700110464947012137028566, −7.32923709637318549938123022578, −6.40064651402118260535471360402, −4.71918010558152361109255365612, −2.57796178341231744450915736756, 0.42466776437871177666053654216, 2.14948212802221181709351710176, 3.51183099180039926440549852796, 6.14787118669008205433950235599, 7.29291721559229598339292201669, 9.611227363159984756231963834077, 10.30015868470689213088133613856, 12.16221577780377593797408970791, 12.67748379491177129885477955509, 13.89169445379660465197029650994

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