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Properties

Label 2-29-29.28-c9-0-6
Degree $2$
Conductor $29$
Sign $-0.466 + 0.884i$
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  + 19.5i·2-s + 267. i·3-s + 129.·4-s − 595.·5-s − 5.24e3·6-s − 893.·7-s + 1.25e4i·8-s − 5.21e4·9-s − 1.16e4i·10-s + 8.49e3i·11-s + 3.47e4i·12-s + 9.60e4·13-s − 1.74e4i·14-s − 1.59e5i·15-s − 1.79e5·16-s − 3.39e5i·17-s + ⋯
L(s)  = 1  + 0.864i·2-s + 1.91i·3-s + 0.253·4-s − 0.426·5-s − 1.65·6-s − 0.140·7-s + 1.08i·8-s − 2.64·9-s − 0.368i·10-s + 0.174i·11-s + 0.483i·12-s + 0.932·13-s − 0.121i·14-s − 0.814i·15-s − 0.682·16-s − 0.986i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.466 + 0.884i$
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: \((2,\ 29,\ (\ :9/2),\ -0.466 + 0.884i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.757562 - 1.25669i\)
\(L(\frac12)\) \(\approx\) \(0.757562 - 1.25669i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-1.77e6 + 3.36e6i)T \)
good2 \( 1 - 19.5iT - 512T^{2} \)
3 \( 1 - 267. iT - 1.96e4T^{2} \)
5 \( 1 + 595.T + 1.95e6T^{2} \)
7 \( 1 + 893.T + 4.03e7T^{2} \)
11 \( 1 - 8.49e3iT - 2.35e9T^{2} \)
13 \( 1 - 9.60e4T + 1.06e10T^{2} \)
17 \( 1 + 3.39e5iT - 1.18e11T^{2} \)
19 \( 1 - 6.64e5iT - 3.22e11T^{2} \)
23 \( 1 - 1.55e6T + 1.80e12T^{2} \)
31 \( 1 - 5.82e6iT - 2.64e13T^{2} \)
37 \( 1 + 9.78e6iT - 1.29e14T^{2} \)
41 \( 1 - 5.66e6iT - 3.27e14T^{2} \)
43 \( 1 - 3.45e7iT - 5.02e14T^{2} \)
47 \( 1 - 2.16e7iT - 1.11e15T^{2} \)
53 \( 1 + 2.53e7T + 3.29e15T^{2} \)
59 \( 1 + 4.31e7T + 8.66e15T^{2} \)
61 \( 1 + 2.77e7iT - 1.16e16T^{2} \)
67 \( 1 + 3.27e8T + 2.72e16T^{2} \)
71 \( 1 - 1.51e8T + 4.58e16T^{2} \)
73 \( 1 + 7.12e7iT - 5.88e16T^{2} \)
79 \( 1 - 5.79e7iT - 1.19e17T^{2} \)
83 \( 1 - 4.22e8T + 1.86e17T^{2} \)
89 \( 1 + 8.21e7iT - 3.50e17T^{2} \)
97 \( 1 - 1.37e9iT - 7.60e17T^{2} \)
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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

−15.99574407514742028998653715175, −15.06983236112990910488329730508, −14.12393290344270860017780869882, −11.65089090804212278064513549201, −10.71527085490760173569407693990, −9.329401344164071684583210972772, −8.035998169483716804347009566854, −6.05783425842141917156677723275, −4.73585741968974480675256509734, −3.21926427419281731273485813474, 0.58364126998095078976195892438, 1.76421209013178727365689093518, 3.18489005826696147461671074777, 6.22479315584278588112824750376, 7.26641461133958504732402208023, 8.654428541442689170802755974941, 10.95007773181463964302922765589, 11.77264763387263259875529086117, 12.83864851523769432813302534795, 13.54836447834610107821469594777

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