Properties

Degree 1
Conductor 83
Sign $0.0614 + 0.998i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.606 + 0.795i)2-s + (−0.973 − 0.227i)3-s + (−0.264 + 0.964i)4-s + (0.953 − 0.301i)5-s + (−0.409 − 0.912i)6-s + (0.477 + 0.878i)7-s + (−0.927 + 0.373i)8-s + (0.896 + 0.443i)9-s + (0.817 + 0.575i)10-s + (−0.997 − 0.0765i)11-s + (0.477 − 0.878i)12-s + (−0.114 + 0.993i)13-s + (−0.409 + 0.912i)14-s + (−0.997 + 0.0765i)15-s + (−0.859 − 0.511i)16-s + (0.338 + 0.941i)17-s + ⋯
L(s,χ)  = 1  + (0.606 + 0.795i)2-s + (−0.973 − 0.227i)3-s + (−0.264 + 0.964i)4-s + (0.953 − 0.301i)5-s + (−0.409 − 0.912i)6-s + (0.477 + 0.878i)7-s + (−0.927 + 0.373i)8-s + (0.896 + 0.443i)9-s + (0.817 + 0.575i)10-s + (−0.997 − 0.0765i)11-s + (0.477 − 0.878i)12-s + (−0.114 + 0.993i)13-s + (−0.409 + 0.912i)14-s + (−0.997 + 0.0765i)15-s + (−0.859 − 0.511i)16-s + (0.338 + 0.941i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0614 + 0.998i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0614 + 0.998i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.0614 + 0.998i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (29, \cdot )$
Sato-Tate  :  $\mu(41)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (0:\ ),\ 0.0614 + 0.998i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7895317864 + 0.7424470195i$
$L(\frac12,\chi)$  $\approx$  $0.7895317864 + 0.7424470195i$
$L(\chi,1)$  $\approx$  0.9975911307 + 0.5587075603i
$L(1,\chi)$  $\approx$  0.9975911307 + 0.5587075603i

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

−30.251797905340018083881929299904, −29.462291903563018525996423041, −28.95155209956143248204349069667, −27.61720691064525064015142028684, −26.80969194611160712499113519319, −25.074047212236709390142246190223, −23.72658419651706267111894692741, −22.99227146183413127886806869791, −22.00464033271798999248724724750, −21.03571397662021660598692607091, −20.25484137897232134236280010858, −18.292091484876042174503375088988, −17.89626303120461330094831512461, −16.40108277795064491158493016353, −14.93338848867777083170902646402, −13.6834795394652609576263513851, −12.780895902144381518744266294629, −11.35505675975029442551953972872, −10.42466777647082708474971885151, −9.76542353764776577502103838723, −7.26811552572106525269699567040, −5.63515958486598203822476256850, −4.99115801727501699782245045092, −3.210851053244489387292216241540, −1.311350878960791353025002364376, 2.234323867643562009160844540027, 4.678330477860464171132922246865, 5.552763873224556349896920833574, 6.45345789679621682681852938987, 7.97951916178385709459947103137, 9.43372768223609974378785607145, 11.17096703130684239573885474189, 12.43151932988415095673816570485, 13.23820958151084314419460362281, 14.554419590122497685574797987190, 15.890153782450281360098695741233, 16.83687880334145824636311627050, 17.831298174808337759780401068279, 18.59689530139430400303656169381, 21.08839164223398411615772710109, 21.567563140947550479887127869938, 22.565322406625766258922143462066, 23.98476692854364021882195667726, 24.34507404811009144085575946877, 25.57333917358622361013257952252, 26.6698865092407965315231191865, 28.26773268740280062298081096118, 28.880170642231846366995121421165, 30.218251237693191163283225237721, 31.17328133797584968157802220727

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