Properties

Degree 1
Conductor 29
Sign $0.694 + 0.719i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.694 + 0.719i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.694 + 0.719i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.694 + 0.719i$
motivic weight  =  \(0\)
character  :  $\chi_{29} (2, \cdot )$
Sato-Tate  :  $\mu(28)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 29,\ (1:\ ),\ 0.694 + 0.719i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.336482735 + 0.9923830322i$
$L(\frac12,\chi)$  $\approx$  $2.336482735 + 0.9923830322i$
$L(\chi,1)$  $\approx$  1.863111678 + 0.5674444185i
$L(1,\chi)$  $\approx$  1.863111678 + 0.5674444185i

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

−37.04245994873128389789044407383, −35.53328136868934734040255487507, −34.24374579899933364274784294576, −32.92354006283801009079104039537, −31.686679141284306475076800817427, −30.40092533147660733581688086032, −29.83366549478286309071852784850, −28.65809470427005415901725935009, −26.20135698442904782383539552225, −25.35571381026268010888032397003, −23.88099632625181502635767661730, −22.82998422810975892378699426484, −21.666462301609682914745859493224, −19.75980147859914222535972092754, −19.20336194434541213821538709993, −17.26736173269111935368574092297, −15.13933593332879386040185508097, −14.08824240930443559458423822691, −12.972308637590797923695844339365, −11.59886860374900127592551732455, −9.81363349136584420519654473246, −7.11863534705811442957686118495, −6.387389263564517957755702031095, −3.69799039299330378979350157956, −2.1340826376337406449548185788, 2.955261698120846657748266819765, 4.57621626133495754081804908699, 5.98716994728022731816997731832, 8.38030389475862952098068207369, 9.914179868587829956569145831799, 11.9449356961725847847431895138, 13.29217724059807264511985439907, 14.66857555159666668385681280792, 16.05859785475565978352102140741, 16.77487183832076024476457739420, 19.627794749167200060024001624399, 20.63467801899391642935435791516, 21.83840967026622279248960708327, 22.77741060955489021162623888160, 24.75193013124266946574827913743, 25.29240827090451953382618167494, 26.979572624964354316361891481803, 28.50198626223536218476637900861, 29.74818009913736294213063516409, 31.60940663267064007496181314545, 32.13373860933532408111793833039, 32.93999228210607712199348630910, 34.3737230564112118469584964880, 35.79909569593469150047336538347, 37.52829101282501412682340320577

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