Properties

Degree 1
Conductor 83
Sign $0.759 - 0.651i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.988 + 0.152i)2-s + (0.0383 − 0.999i)3-s + (0.953 + 0.301i)4-s + (−0.543 − 0.839i)5-s + (0.190 − 0.981i)6-s + (0.338 + 0.941i)7-s + (0.896 + 0.443i)8-s + (−0.997 − 0.0765i)9-s + (−0.409 − 0.912i)10-s + (−0.859 − 0.511i)11-s + (0.338 − 0.941i)12-s + (0.720 − 0.693i)13-s + (0.190 + 0.981i)14-s + (−0.859 + 0.511i)15-s + (0.817 + 0.575i)16-s + (−0.665 + 0.746i)17-s + ⋯
L(s,χ)  = 1  + (0.988 + 0.152i)2-s + (0.0383 − 0.999i)3-s + (0.953 + 0.301i)4-s + (−0.543 − 0.839i)5-s + (0.190 − 0.981i)6-s + (0.338 + 0.941i)7-s + (0.896 + 0.443i)8-s + (−0.997 − 0.0765i)9-s + (−0.409 − 0.912i)10-s + (−0.859 − 0.511i)11-s + (0.338 − 0.941i)12-s + (0.720 − 0.693i)13-s + (0.190 + 0.981i)14-s + (−0.859 + 0.511i)15-s + (0.817 + 0.575i)16-s + (−0.665 + 0.746i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.759 - 0.651i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.759 - 0.651i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.759 - 0.651i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (4, \cdot )$
Sato-Tate  :  $\mu(41)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (0:\ ),\ 0.759 - 0.651i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.479267430 - 0.5474772023i$
$L(\frac12,\chi)$  $\approx$  $1.479267430 - 0.5474772023i$
$L(\chi,1)$  $\approx$  1.560063876 - 0.3738821907i
$L(1,\chi)$  $\approx$  1.560063876 - 0.3738821907i

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

−31.01499119356857955299324003106, −30.32596477958914174997634200691, −28.94870123287652824485674636766, −27.92556048743298247804539581506, −26.43350533052393000321295354348, −26.084721715258716841797297221728, −24.23456832864036386869331664207, −23.108250077917351051955271343145, −22.599427271593828626169547213831, −21.306490305998828502424411268350, −20.48817245798705809916531996082, −19.5423524813231090520027150669, −17.831741893037537287475431308747, −16.1229718767728365026191440831, −15.56818561812585859507644357247, −14.338928013808495299689105407775, −13.57332760651572414932579921429, −11.6201126265342782763873230237, −10.94269341381814825082821324500, −9.93429798300219437969562612003, −7.806380734519246146777197841855, −6.54063113680052725115699522046, −4.807662098246490972963196481307, −3.95425244162488348585409335152, −2.6550867649065598736766105924, 1.80443109893056413378980024669, 3.38742415724186439137773878052, 5.234290068822049768744203035, 6.06997908512940945938670466691, 7.85442576254966617491327890326, 8.4491633877780824062597268080, 11.03873717133513308462856324693, 12.12974177657787642140341701147, 12.836156570257773594322270540131, 13.85516060949972597800755034257, 15.30169193709993051623217769849, 16.16002044427349629991231694899, 17.65545760466743444488910111357, 18.91132897722617546963798149416, 20.138261121018011261088881692484, 21.00122006836674972040037177318, 22.38127156819892527284869312715, 23.54414925017461772419075773759, 24.24976789086925166564549953114, 24.944009924188600804350173726215, 26.07505337034195494755322004249, 28.0162496745319578590580106467, 28.80421300746352699700910698370, 29.91391273502495043625123777489, 31.19340969297100560320933123460

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