Properties

Degree 1
Conductor 29
Sign $0.748 + 0.663i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.748 + 0.663i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.748 + 0.663i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.748 + 0.663i$
motivic weight  =  \(0\)
character  :  $\chi_{29} (21, \cdot )$
Sato-Tate  :  $\mu(28)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 29,\ (1:\ ),\ 0.748 + 0.663i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.087081239 + 0.4123738032i$
$L(\frac12,\chi)$  $\approx$  $1.087081239 + 0.4123738032i$
$L(\chi,1)$  $\approx$  0.9517205724 + 0.1276036800i
$L(1,\chi)$  $\approx$  0.9517205724 + 0.1276036800i

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

−36.50020029788882475712826055811, −35.844377000987124402476042857646, −34.78843057746584802474589897246, −32.79690552463707388910807018314, −32.341501795118824234035018108792, −30.696356685737118594815641196403, −29.208314400232032490220014496006, −27.65574336461099084076831662253, −26.63842281989049923602423545720, −25.60382161244306762187339201257, −24.22176112881330084964965045254, −23.525010012698904818179661474238, −20.86140831918805662796470377969, −19.70135093836385893835531792831, −18.944522888903973775404159782323, −17.0044815211619784788108709693, −15.954688811106935113934370831995, −14.48174747157975039406996363439, −13.119715830924541798111648536269, −10.82428153570607947611710280105, −8.99537501336799823116965325086, −8.15878852533203210536033250729, −6.63009378732893884977654244674, −4.032093181053488653242247587450, −1.108006771234075177172717499301, 2.3662445182461734522183289831, 3.746600394393176169284577965833, 7.15117898661048565153829143194, 8.55162982325935521455889507174, 9.785296740604278280112876041834, 11.348964611897701878111886519811, 12.88744644711863407398966621349, 14.89116771884443934774591593765, 15.940840478562203687453697352655, 18.13045620199102475244654139509, 19.04373480829460525105877112993, 20.122142224187477977562533148425, 21.40393052872404783201839018658, 22.72615278105629039279741289762, 25.18560929464432528653516989071, 25.782990453369060922239324232262, 27.21136163695190757224101260428, 27.97443704409467097341639975847, 29.826163799698810649399415439403, 30.86154201128395854661805431599, 31.64937470322825613604807963353, 33.61056647325815667431121405335, 35.08979317042307321814473340728, 35.94063821357781366743636405717, 37.46986994404126658566257533250

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