Properties

Degree 1
Conductor 29
Sign $0.976 + 0.214i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.976 + 0.214i$
motivic weight  =  \(0\)
character  :  $\chi_{29} (13, \cdot )$
Sato-Tate  :  $\mu(14)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 29,\ (0:\ ),\ 0.976 + 0.214i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6083883414 + 0.06602628762i$
$L(\frac12,\chi)$  $\approx$  $0.6083883414 + 0.06602628762i$
$L(\chi,1)$  $\approx$  0.7883965330 + 0.02553417020i
$L(1,\chi)$  $\approx$  0.7883965330 + 0.02553417020i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

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