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Properties

Label 2-29-29.6-c1-0-1
Degree $2$
Conductor $29$
Sign $0.910 - 0.414i$
Analytic cond. $0.231566$
Root an. cond. $0.481213$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.21 + 0.965i)2-s + (−2.86 − 0.653i)3-s + (0.0890 + 0.390i)4-s + (0.283 − 0.355i)5-s + (−2.83 − 3.55i)6-s + (−0.759 + 3.32i)7-s + (1.07 − 2.23i)8-s + (5.05 + 2.43i)9-s + (0.686 − 0.156i)10-s + (−0.635 − 1.31i)11-s − 1.17i·12-s + (−2.78 + 1.33i)13-s + (−4.13 + 3.29i)14-s + (−1.04 + 0.831i)15-s + (4.18 − 2.01i)16-s − 2.82i·17-s + ⋯
L(s)  = 1  + (0.856 + 0.683i)2-s + (−1.65 − 0.377i)3-s + (0.0445 + 0.195i)4-s + (0.126 − 0.158i)5-s + (−1.15 − 1.45i)6-s + (−0.287 + 1.25i)7-s + (0.380 − 0.789i)8-s + (1.68 + 0.811i)9-s + (0.217 − 0.0495i)10-s + (−0.191 − 0.397i)11-s − 0.339i·12-s + (−0.771 + 0.371i)13-s + (−1.10 + 0.881i)14-s + (−0.269 + 0.214i)15-s + (1.04 − 0.503i)16-s − 0.684i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.910 - 0.414i$
Analytic conductor: \(0.231566\)
Root analytic conductor: \(0.481213\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :1/2),\ 0.910 - 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.668586 + 0.145126i\)
\(L(\frac12)\) \(\approx\) \(0.668586 + 0.145126i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (3.83 + 3.77i)T \)
good2 \( 1 + (-1.21 - 0.965i)T + (0.445 + 1.94i)T^{2} \)
3 \( 1 + (2.86 + 0.653i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (-0.283 + 0.355i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (0.759 - 3.32i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (0.635 + 1.31i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (2.78 - 1.33i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + (-2.23 + 0.510i)T + (17.1 - 8.24i)T^{2} \)
23 \( 1 + (-0.333 - 0.417i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (2.80 + 2.23i)T + (6.89 + 30.2i)T^{2} \)
37 \( 1 + (3.08 - 6.41i)T + (-23.0 - 28.9i)T^{2} \)
41 \( 1 - 4.43iT - 41T^{2} \)
43 \( 1 + (-3.22 + 2.57i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (-0.0778 - 0.161i)T + (-29.3 + 36.7i)T^{2} \)
53 \( 1 + (-1.92 + 2.41i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 + (-1.84 - 0.420i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (10.9 + 5.28i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (-4.06 + 1.95i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-6.47 + 5.15i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + (6.00 - 12.4i)T + (-49.2 - 61.7i)T^{2} \)
83 \( 1 + (-0.545 - 2.39i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (10.5 + 8.41i)T + (19.8 + 86.7i)T^{2} \)
97 \( 1 + (-4.58 + 1.04i)T + (87.3 - 42.0i)T^{2} \)
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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

−16.96300039835462278141304608818, −16.09449698368139733424307811940, −15.08368853952578303993357095330, −13.42451447167535721349930674859, −12.38958063914186608454362589801, −11.42024959805681677350971713369, −9.660306036899778590060541711702, −7.08283534664582326321460632384, −5.82135922482034502991718014869, −5.08858261477480530779716134548, 4.08029133751381065579970008725, 5.40093903660652915883022383970, 7.18870989812725013803472524124, 10.22722885867857454948365047783, 10.86229233904019356269750180357, 12.16471509821507770596997800696, 12.98039160357999460067384584742, 14.45458683535821128204224703639, 16.21216265700384355528052555231, 17.12469549048324073726820248930

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