Properties

Degree 1
Conductor 29
Sign $0.226 - 0.974i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.226 - 0.974i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.226 - 0.974i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.226 - 0.974i$
motivic weight  =  \(0\)
character  :  $\chi_{29} (25, \cdot )$
Sato-Tate  :  $\mu(7)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 29,\ (0:\ ),\ 0.226 - 0.974i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4416425005 - 0.3507984638i$
$L(\frac12,\chi)$  $\approx$  $0.4416425005 - 0.3507984638i$
$L(\chi,1)$  $\approx$  0.6528919956 - 0.3316007111i
$L(1,\chi)$  $\approx$  0.6528919956 - 0.3316007111i

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

−37.5323488185379158211807198535, −36.56188202429477111490678345390, −34.71787583097152859425920222862, −34.33401996939299147590523320097, −32.65432489155710833939638683109, −31.590305978208988471360313284115, −30.13141659567623261227008763392, −28.13496151293009831090829527658, −27.3436392338942464677126124544, −26.461211388344044000001809037043, −25.17395709814187193342625292711, −23.931607996088498172389149957563, −22.08842207739531041879063235733, −20.523860204505413932175369198663, −19.3478096511209205288240911622, −18.216148454145388910059337790068, −16.30709588189565136484897237965, −15.36017234408214295049683752158, −14.39213514394595945283508975255, −11.54483671365899722668504897032, −10.31874047389810484869640923239, −8.63067405279540414287251963179, −7.75778970490669906986479694745, −5.37988265927970739855307872873, −2.96022208089554177309983676683, 1.61214558537285470986398597683, 3.848044766361780880431196628190, 7.313529678744304157539183855230, 7.97260225557239053948840210802, 9.61514612098094907729874945200, 11.57291529803550573585487992672, 12.60332597562818511398220771882, 14.48346948435541565153666979902, 16.31266312757757809585291694314, 17.73667931611430426524965741514, 19.00574120008196987920529678340, 20.09336404722979854887021504269, 20.86265942832055770752567184658, 23.361976142751984272507738870114, 24.458339000920987713214654022760, 25.87411516241852202531642179144, 26.953794204936854909656695631642, 28.179654925459393307074156217980, 29.59550062923938404567594411412, 30.68910156125391744068091843285, 31.52394370920901009923278116535, 33.66021342719269686084723334747, 35.05130917875897870589104782770, 36.19397129286534619407451083881, 36.54453494300704712246551213559

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