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Properties

Label 2-29-29.28-c9-0-16
Degree $2$
Conductor $29$
Sign $-0.567 + 0.823i$
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  − 27.8i·2-s − 61.9i·3-s − 262.·4-s + 1.30e3·5-s − 1.72e3·6-s + 8.29e3·7-s − 6.95e3i·8-s + 1.58e4·9-s − 3.62e4i·10-s − 3.44e3i·11-s + 1.62e4i·12-s + 7.66e4·13-s − 2.30e5i·14-s − 8.05e4i·15-s − 3.27e5·16-s + 3.81e5i·17-s + ⋯
L(s)  = 1  − 1.22i·2-s − 0.441i·3-s − 0.511·4-s + 0.931·5-s − 0.542·6-s + 1.30·7-s − 0.600i·8-s + 0.805·9-s − 1.14i·10-s − 0.0710i·11-s + 0.225i·12-s + 0.744·13-s − 1.60i·14-s − 0.410i·15-s − 1.24·16-s + 1.10i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.567 + 0.823i$
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: \((2,\ 29,\ (\ :9/2),\ -0.567 + 0.823i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.25118 - 2.38119i\)
\(L(\frac12)\) \(\approx\) \(1.25118 - 2.38119i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-2.16e6 + 3.13e6i)T \)
good2 \( 1 + 27.8iT - 512T^{2} \)
3 \( 1 + 61.9iT - 1.96e4T^{2} \)
5 \( 1 - 1.30e3T + 1.95e6T^{2} \)
7 \( 1 - 8.29e3T + 4.03e7T^{2} \)
11 \( 1 + 3.44e3iT - 2.35e9T^{2} \)
13 \( 1 - 7.66e4T + 1.06e10T^{2} \)
17 \( 1 - 3.81e5iT - 1.18e11T^{2} \)
19 \( 1 - 5.40e4iT - 3.22e11T^{2} \)
23 \( 1 + 2.27e6T + 1.80e12T^{2} \)
31 \( 1 - 2.48e6iT - 2.64e13T^{2} \)
37 \( 1 + 7.46e6iT - 1.29e14T^{2} \)
41 \( 1 - 2.80e7iT - 3.27e14T^{2} \)
43 \( 1 + 2.67e7iT - 5.02e14T^{2} \)
47 \( 1 - 5.19e7iT - 1.11e15T^{2} \)
53 \( 1 + 3.40e7T + 3.29e15T^{2} \)
59 \( 1 - 7.09e7T + 8.66e15T^{2} \)
61 \( 1 + 1.40e8iT - 1.16e16T^{2} \)
67 \( 1 + 3.74e7T + 2.72e16T^{2} \)
71 \( 1 + 3.17e8T + 4.58e16T^{2} \)
73 \( 1 + 2.89e7iT - 5.88e16T^{2} \)
79 \( 1 - 3.43e8iT - 1.19e17T^{2} \)
83 \( 1 + 4.04e8T + 1.86e17T^{2} \)
89 \( 1 - 6.56e8iT - 3.50e17T^{2} \)
97 \( 1 - 7.43e8iT - 7.60e17T^{2} \)
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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

−14.15139387055674317931097361240, −13.15058615956159307280495672612, −12.03158635042264741728692418209, −10.81848079961107382927467209487, −9.835094554171936302067063681144, −8.079328602591250703315400513732, −6.19699055181216400741355896281, −4.16824595786665709630583930982, −2.01901479001750055125406751391, −1.33791380230635868079354397706, 1.81116294563180301594364935575, 4.63034960994651343583598711697, 5.81447594567159680759436755419, 7.31003931868194501943205105455, 8.631848228847861800619130402385, 10.12847744603611362563140148069, 11.56864423633736350617696234765, 13.61419847439939921796270447805, 14.40876779320658973978986683281, 15.58096476891495134118251468743

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