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Properties

Label 2-29-29.12-c8-0-0
Degree $2$
Conductor $29$
Sign $0.795 - 0.605i$
Analytic cond. $11.8139$
Root an. cond. $3.43714$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.63 − 6.63i)2-s + (−21.3 − 21.3i)3-s − 167. i·4-s + 656. i·5-s + 283. i·6-s − 2.10e3·7-s + (−2.81e3 + 2.81e3i)8-s − 5.64e3i·9-s + (4.35e3 − 4.35e3i)10-s + (1.46e4 + 1.46e4i)11-s + (−3.58e3 + 3.58e3i)12-s + 863. i·13-s + (1.39e4 + 1.39e4i)14-s + (1.40e4 − 1.40e4i)15-s − 5.69e3·16-s + (8.89e4 + 8.89e4i)17-s + ⋯
L(s)  = 1  + (−0.414 − 0.414i)2-s + (−0.263 − 0.263i)3-s − 0.656i·4-s + 1.04i·5-s + 0.218i·6-s − 0.875·7-s + (−0.686 + 0.686i)8-s − 0.860i·9-s + (0.435 − 0.435i)10-s + (0.997 + 0.997i)11-s + (−0.173 + 0.173i)12-s + 0.0302i·13-s + (0.363 + 0.363i)14-s + (0.276 − 0.276i)15-s − 0.0868·16-s + (1.06 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.795 - 0.605i$
Analytic conductor: \(11.8139\)
Root analytic conductor: \(3.43714\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :4),\ 0.795 - 0.605i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.877249 + 0.295846i\)
\(L(\frac12)\) \(\approx\) \(0.877249 + 0.295846i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (6.33e5 - 3.14e5i)T \)
good2 \( 1 + (6.63 + 6.63i)T + 256iT^{2} \)
3 \( 1 + (21.3 + 21.3i)T + 6.56e3iT^{2} \)
5 \( 1 - 656. iT - 3.90e5T^{2} \)
7 \( 1 + 2.10e3T + 5.76e6T^{2} \)
11 \( 1 + (-1.46e4 - 1.46e4i)T + 2.14e8iT^{2} \)
13 \( 1 - 863. iT - 8.15e8T^{2} \)
17 \( 1 + (-8.89e4 - 8.89e4i)T + 6.97e9iT^{2} \)
19 \( 1 + (-6.95e4 - 6.95e4i)T + 1.69e10iT^{2} \)
23 \( 1 - 2.75e5T + 7.83e10T^{2} \)
31 \( 1 + (5.95e5 + 5.95e5i)T + 8.52e11iT^{2} \)
37 \( 1 + (7.99e5 - 7.99e5i)T - 3.51e12iT^{2} \)
41 \( 1 + (1.70e6 - 1.70e6i)T - 7.98e12iT^{2} \)
43 \( 1 + (-1.55e6 - 1.55e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (5.21e6 - 5.21e6i)T - 2.38e13iT^{2} \)
53 \( 1 - 1.26e7T + 6.22e13T^{2} \)
59 \( 1 + 1.33e7T + 1.46e14T^{2} \)
61 \( 1 + (-1.25e7 - 1.25e7i)T + 1.91e14iT^{2} \)
67 \( 1 - 2.41e7iT - 4.06e14T^{2} \)
71 \( 1 + 1.87e7iT - 6.45e14T^{2} \)
73 \( 1 + (-2.71e7 + 2.71e7i)T - 8.06e14iT^{2} \)
79 \( 1 + (-8.80e6 - 8.80e6i)T + 1.51e15iT^{2} \)
83 \( 1 + 5.50e7T + 2.25e15T^{2} \)
89 \( 1 + (4.69e7 + 4.69e7i)T + 3.93e15iT^{2} \)
97 \( 1 + (-2.51e6 + 2.51e6i)T - 7.83e15iT^{2} \)
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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

−14.98789934880470357310996846451, −14.60726915455710922317537503298, −12.64432612802242165258006070997, −11.48830115386139079339732603564, −10.15954017238060819106712279053, −9.338903899925097068007497364433, −6.98579993972055820527688242280, −6.00154140358414251961874675039, −3.36028479151510967676890356273, −1.36729448793705305732757363524, 0.53842581667387614257593937815, 3.43885348946965050743150282943, 5.32981368827946891962004809426, 7.11149380765904662227546065927, 8.631222831609168698981478042700, 9.537200878079225996151372257785, 11.47576764368784301183689244444, 12.66854045755652539612364934281, 13.75291173114943229178245617513, 15.80572292111624390893097166759

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