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Properties

Label 2-29-29.17-c10-0-15
Degree $2$
Conductor $29$
Sign $0.988 + 0.150i$
Analytic cond. $18.4253$
Root an. cond. $4.29247$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−16.1 + 16.1i)2-s + (130. − 130. i)3-s + 505. i·4-s − 1.79e3i·5-s + 4.19e3i·6-s + 1.41e4·7-s + (−2.46e4 − 2.46e4i)8-s + 2.51e4i·9-s + (2.88e4 + 2.88e4i)10-s + (4.40e3 − 4.40e3i)11-s + (6.57e4 + 6.57e4i)12-s − 5.44e5i·13-s + (−2.27e5 + 2.27e5i)14-s + (−2.33e5 − 2.33e5i)15-s + 2.75e5·16-s + (−9.96e5 + 9.96e5i)17-s + ⋯
L(s)  = 1  + (−0.503 + 0.503i)2-s + (0.535 − 0.535i)3-s + 0.493i·4-s − 0.573i·5-s + 0.538i·6-s + 0.841·7-s + (−0.751 − 0.751i)8-s + 0.426i·9-s + (0.288 + 0.288i)10-s + (0.0273 − 0.0273i)11-s + (0.264 + 0.264i)12-s − 1.46i·13-s + (−0.423 + 0.423i)14-s + (−0.307 − 0.307i)15-s + 0.262·16-s + (−0.701 + 0.701i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.988 + 0.150i$
Analytic conductor: \(18.4253\)
Root analytic conductor: \(4.29247\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :5),\ 0.988 + 0.150i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.82754 - 0.138089i\)
\(L(\frac12)\) \(\approx\) \(1.82754 - 0.138089i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-2.04e7 + 8.08e5i)T \)
good2 \( 1 + (16.1 - 16.1i)T - 1.02e3iT^{2} \)
3 \( 1 + (-130. + 130. i)T - 5.90e4iT^{2} \)
5 \( 1 + 1.79e3iT - 9.76e6T^{2} \)
7 \( 1 - 1.41e4T + 2.82e8T^{2} \)
11 \( 1 + (-4.40e3 + 4.40e3i)T - 2.59e10iT^{2} \)
13 \( 1 + 5.44e5iT - 1.37e11T^{2} \)
17 \( 1 + (9.96e5 - 9.96e5i)T - 2.01e12iT^{2} \)
19 \( 1 + (-2.32e6 + 2.32e6i)T - 6.13e12iT^{2} \)
23 \( 1 - 1.18e7T + 4.14e13T^{2} \)
31 \( 1 + (-3.56e6 + 3.56e6i)T - 8.19e14iT^{2} \)
37 \( 1 + (7.58e7 + 7.58e7i)T + 4.80e15iT^{2} \)
41 \( 1 + (-1.06e8 - 1.06e8i)T + 1.34e16iT^{2} \)
43 \( 1 + (-9.66e7 + 9.66e7i)T - 2.16e16iT^{2} \)
47 \( 1 + (1.41e8 + 1.41e8i)T + 5.25e16iT^{2} \)
53 \( 1 + 2.71e8T + 1.74e17T^{2} \)
59 \( 1 - 1.00e9T + 5.11e17T^{2} \)
61 \( 1 + (6.62e8 - 6.62e8i)T - 7.13e17iT^{2} \)
67 \( 1 + 1.65e9iT - 1.82e18T^{2} \)
71 \( 1 - 1.90e9iT - 3.25e18T^{2} \)
73 \( 1 + (1.11e9 + 1.11e9i)T + 4.29e18iT^{2} \)
79 \( 1 + (-1.79e8 + 1.79e8i)T - 9.46e18iT^{2} \)
83 \( 1 + 2.08e8T + 1.55e19T^{2} \)
89 \( 1 + (1.14e9 - 1.14e9i)T - 3.11e19iT^{2} \)
97 \( 1 + (-5.12e9 - 5.12e9i)T + 7.37e19iT^{2} \)
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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

−14.94582269630804747405135321512, −13.36197808463832763855431735205, −12.59791809326260424202521994973, −10.92110813350117538441949997792, −8.915864204076345368749453540961, −8.156400083791400057958275630886, −7.13371272481305943484227327870, −5.01563943704324637715668743471, −2.86485758736368849156434714972, −0.921178803693969735647070626466, 1.28264061461337209007549557924, 2.88361026794694597996930572396, 4.77181058685386263438251459311, 6.75037749570317962274636677276, 8.731899944632452923241095627654, 9.599628579882841949227536716950, 10.89853290066868673167127954432, 11.81464805605574923440361363691, 14.15324890325934645934162811050, 14.59525819111862439556632878413

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