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Properties

Label 1-29-29.13-r0-0-0
Degree $1$
Conductor $29$
Sign $0.976 + 0.214i$
Analytic cond. $0.134675$
Root an. cond. $0.134675$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.623 − 0.781i)2-s + (0.222 + 0.974i)3-s + (−0.222 + 0.974i)4-s + (0.623 + 0.781i)5-s + (0.623 − 0.781i)6-s + (−0.222 − 0.974i)7-s + (0.900 − 0.433i)8-s + (−0.900 + 0.433i)9-s + (0.222 − 0.974i)10-s + (0.900 + 0.433i)11-s − 12-s + (−0.900 − 0.433i)13-s + (−0.623 + 0.781i)14-s + (−0.623 + 0.781i)15-s + (−0.900 − 0.433i)16-s − 17-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)2-s + (0.222 + 0.974i)3-s + (−0.222 + 0.974i)4-s + (0.623 + 0.781i)5-s + (0.623 − 0.781i)6-s + (−0.222 − 0.974i)7-s + (0.900 − 0.433i)8-s + (−0.900 + 0.433i)9-s + (0.222 − 0.974i)10-s + (0.900 + 0.433i)11-s − 12-s + (−0.900 − 0.433i)13-s + (−0.623 + 0.781i)14-s + (−0.623 + 0.781i)15-s + (−0.900 − 0.433i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(29\)
Sign: $0.976 + 0.214i$
Analytic conductor: \(0.134675\)
Root analytic conductor: \(0.134675\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 29,\ (0:\ ),\ 0.976 + 0.214i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6083883414 + 0.06602628762i\)
\(L(\frac12)\) \(\approx\) \(0.6083883414 + 0.06602628762i\)
\(L(1)\) \(\approx\) \(0.7883965330 + 0.02553417020i\)
\(L(1)\) \(\approx\) \(0.7883965330 + 0.02553417020i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
good2 \( 1 + (-0.623 - 0.781i)T \)
3 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (0.623 + 0.781i)T \)
7 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 - T \)
19 \( 1 + (0.222 - 0.974i)T \)
23 \( 1 + (0.623 - 0.781i)T \)
31 \( 1 + (-0.623 - 0.781i)T \)
37 \( 1 + (0.900 - 0.433i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 + T \)
61 \( 1 + (0.222 + 0.974i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.623 + 0.781i)T \)
79 \( 1 + (0.900 - 0.433i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (-0.623 - 0.781i)T \)
97 \( 1 + (0.222 - 0.974i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−37.03239018300075955204800713983, −35.77824204303166023109575905467, −35.15274991409048139537864142083, −33.78482971744553534557033182184, −32.305833783520777355026410706, −31.356576843216698526147330032903, −29.29674209458594659369960831860, −28.64121023363275523783230519129, −27.07915767037595543984595556317, −25.32819837525937167939053667351, −24.87667645984231556792229612477, −23.86497942399238150699415325211, −22.060581257340721983034857809513, −19.97928032983757276757401104507, −18.89891789053243359176562381915, −17.653741148250273398151312434009, −16.584205453924954310134803867579, −14.78250135534847991420968381867, −13.47814370132204114869983233527, −11.93187442094132334401638178102, −9.39464273918183388239047316158, −8.5355147329953139271113761391, −6.73786330573043586911787075873, −5.482186346816370177672703500988, −1.79704209862112960798117665156, 2.69742322737733127522640991731, 4.31298654705741667379084489725, 7.10276257426064490164994212848, 9.20488413913909512982043806358, 10.19064034340018371368928321077, 11.22888108526560097890506812715, 13.3787976514636756084575399121, 14.83518533614478487740314286282, 16.78163467592222915551179769535, 17.6718923704005922911898338324, 19.574272209590600554726892202298, 20.40944745169734506315066958017, 21.94904749354369692727562581758, 22.53756123664507711863280761088, 25.26216995339650035849056870236, 26.47004428516592641797358001086, 27.040134788874673755295340287929, 28.54569274569076540073376565845, 29.77332705306454981284737596200, 30.80165329189149909620047304727, 32.50189181403968140300386152824, 33.55543307333167460465839718879, 34.96418285734596761263032118124, 36.6134119837898138381919212424, 37.38114260259130095268915392751

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